# Paul M Cohn's books

Paul Cohn has written many books. Rather than write standard further editions, he favoured repackaging many of his texts dividing them up in different ways. This means that presenting the works in our normal format, where he put second and later editions immediately below the first, is impossible. We have, therefore, simply listed the books, twenty-five in total, in chronological order of publication. We give information about these such as a description by the publisher, an extract from the Preface and extracts from some reviews.

Lie groups (1957)

Linear equations (1958)

Solid geometry (1961)

Universal algebra (1965)

Morita equivalence and duality (1968)

Lectures on algebraic numbers and algebraic functions (1969)

Free rings and their relations (1971)

Skew field constructions (1973)

Algebra Volume 1 (1974)

Skew field constructions (1977)

Algebra Volume 2 (1977)

Universal algebra (Second Edition) (1981)

Algebra Volume 1 (Second Edition) (1982)

Free rings and their relations (Second Edition) (1985)

Algebra Volume 2 (Second Edition) (1989)

Algebra Volume 3 (Second Edition) (1991)

Algebraic numbers and algebraic functions (1991)

Elements of linear algebra (1994)

Skew fields (1995)

Algebra Vols 1, 2 and 3 (Second paperback edition) (1995)

Introduction to ring theory (2000)

Classic algebra (2002)

Basic algebra: groups, rings and fields (2002)

Further algebra and applications (2003)

Free ideal rings and localization in general rings (2006)

**Click on a link below to go to the information about that book**Lie groups (1957)

Linear equations (1958)

Solid geometry (1961)

Universal algebra (1965)

Morita equivalence and duality (1968)

Lectures on algebraic numbers and algebraic functions (1969)

Free rings and their relations (1971)

Skew field constructions (1973)

Algebra Volume 1 (1974)

Skew field constructions (1977)

Algebra Volume 2 (1977)

Universal algebra (Second Edition) (1981)

Algebra Volume 1 (Second Edition) (1982)

Free rings and their relations (Second Edition) (1985)

Algebra Volume 2 (Second Edition) (1989)

Algebra Volume 3 (Second Edition) (1991)

Algebraic numbers and algebraic functions (1991)

Elements of linear algebra (1994)

Skew fields (1995)

Algebra Vols 1, 2 and 3 (Second paperback edition) (1995)

Introduction to ring theory (2000)

Classic algebra (2002)

Basic algebra: groups, rings and fields (2002)

Further algebra and applications (2003)

Free ideal rings and localization in general rings (2006)

**1. Lie groups (1957), by Paul M Cohn.**

**1.1. From the Publisher.**

The theory of Lie groups rests on three pillars: analysis, topology and algebra. Correspondingly it is possible to distinguish several phases, overlapping in some degree, in its development. It also allows one to regard the subject from different points of view, and it is the algebraic standpoint which has been chosen in this tract as the most suitable one for a first introduction to the subject. The aim has been to develop the beginnings of the theory of Lie groups, especially the fundamental theorems of Lie relating the group to its infinitesimal generators (the Lie algebra); this account occupies the first five chapters. Next to Lie's theorems in importance come the basic properties of subgroups and homomorphisms, and they form the content of Chapter VI. The final chapter, on the universal covering group, could perhaps be most easily dispensed with, but, it is hoped, justifies its existence by bringing back into circulation Schreier's elegant method of constructing covering groups.

**1.2. Review by: Hsien Chung Wang.**

*Amer. Math. Monthly*

**65**(8) (1958), 646.

This monograph is intended, as the author mentioned in the preface, to develop the beginning of the theory of Lie groups, especially the fundamental theorems of Lie relating the group to its Lie algebra. The approach is somewhat standard. Nevertheless the author presents the material in a very clear and self-contained manner. Only the most elementary knowledge of groups, vector spaces and general topology is assumed. Among the propositions and definitions, there are illuminating remarks giving the motivation of the concepts and theorems. The organisation is good, and the style is so lucid that it can serve as an excellent introduction to the theory of Lie groups.

**1.3. Review by: Masatake Kuranishi.**

*Mathematical Reviews*MR0103940

**(21 #2702)**.

This is a concise and elementary textbook on the beginnings of the theory of Lie groups, covering the so-called Lie's fundamental theorems. It also includes the theory of universal covering groups. The readers are required to have only elementary knowledge about groups, vector spaces, topology, and analysis. Numbers of examples are inserted to help the readers.

**1.4. Review by: Kurt A Hirsch.**

*The Mathematical Gazette*

**44**(347) (1960), 78-79.

This is a lucidly written introduction to one of the finest and most imposing edifices modern mathematics has erected. It is inherent in the theory of Lie Groups, both historically and systematically, that it requires a heavy apparatus from several branches of mathematics. The first two chapters of the book give an account of this background material from analysis, topology and algebra. They deal with the elements of the theory of analytic manifolds, infinitesimal transformations and their differential forms; they then introduce the fundamental definition of topological groups, local groups, and Lie groups. In Chapter 3 the operation of commutation of infinitesimal transformations is introduced and the Lie algebra associated with a Lie group is constructed as the algebra of infinitesimal right translations of the group. Chapter 4 develops the operations on differential forms and culminates in the derivation of the Maurer-Cartan equations. These are used in Chapter 5 to prove the existence and uniqueness of a local Lie group having a given Lie algebra. (The more recent global aspect of the theory is not included.) The requisite facts from the theory of differential equations are proved in an Appendix. Chapter 6 studies the connections between subgroups (subalgebras) and homomorphisms of Lie groups (algebras). In particular, it is proved that every continuous homomorphism of a Lie group is analytic and that every closed subgroup of a Lie group is itself a Lie group. The final chapter contains the construction of a universal covering group for every connected Lie group.

The reader of this slim book of 160 pages is well equipped to tackle one of the more detailed accounts of our present knowledge of Lie groups such as Chevalley's well-known three volumes which, of course, go considerably further in depth and width.

**1.5. Review by: A Nijenhuis.**

*Bull. Amer. Math. Soc*.

**65**(1959), 338-341.

It is only in recent years that the study of Lie groups as global objects has come to be considered a subject of general interest. Until that time, it was customary to study only the "group germ" of a group which, upon analysis of the best-known literature, was some unspecified neighbourhood of the identity element of the group. Among the familiar works along this line we find Eisenhart's book on continuous groups, while a quite recent discussion of the same nature is found in a chapter of Schouten's Ricci calculus (second edition).

At quite an early date, Elie Cartan was aware of the global nature of Lie groups; of connectedness and compactness and such; but his work never found a really large audience. This was partly due to his incomplete proofs, but mainly to the difficulty of understanding his general terminology and machinery.

The one text that strongly promoted the interest in Lie groups as global objects was Chevalley's well-known book in the Princeton series. Not only did Chevalley give a coherent and logically complete description of the general nature of Lie groups; he also gave it in a language which could be understood much more readily than Cartan's. In fact, this one book has done more than that: it has stimulated great interest in an "intrinsic" ( = coordinate-free) approach to differential geometry; and it is still frequently quoted in research publications as a general reference for the methods.

Thus, Chevalley's text achieved three things at once: (i) it treated Lie groups as global objects; (ii) it was rigorous; (iii) it had a "new language." While the first two items are generally approved of as positive gains, the third is of a more controversial nature. The modern differential geometer is very happy about it - by definition! - the topologist and algebraist also prefer mappings, linear functionals and such; but the classical differential geometer and the (much more numerous) analyst may very well find that they get too much of too many different kinds of formalism at once. After all, aren't Lie groups essentially just groups whose elements can be described by a number of parameters, while the group operations are expressed by analytic functions of these parameters? In Chevalley, he frequently cannot find the parameters (co-ordinates) or the functions. For somewhat similar reasons, a student with an adequate knowledge of analysis, but little mathematical sophistication may easily find himself snowed under.

It seems that Cohn wrote his little booklet, here under review, with just these latter groups of people in mind. He has made a serious and quite successful effort to preserve the full advantage of the first two objectives of Chevalley: global treatment and rigour; but he has "softened up" on the language of presentation. While any specialised language with clever possibilities - whether "classical" or "intrinsic" - can be mastered only after considerable practice, the author has tried to do away with anything that cannot be considered a straight- forward expression in truly simple terminology. As a consequence, few results "come tumbling out" (which is only possible after creating sophisticated tools in some previous chapter), but proofs are of reasonable length and in each instance the reader sees quite clearly what is going on. As a result, we have here a book which easily lends itself to a course (it will particularly appeal to a non-specialist) or better even, a seminar in which inexperienced students can actively participate.

**2. Linear equations (1958), by Paul M Cohn.**

**2.1. From the Publisher.**

Linear equations play an important part, not only in mathematics itself, but also in many fields in which mathematics is used. Whether we deal with elastic deformations or electrical networks, the flutter of aeroplane wings or the estimation of errors by the method of least squares, at some stage in the calculation we encounter a system of linear equations. In each case the problem of solving the equations is the same, and it is with the mathematical treatment of this question that this book is concerned. By meeting the problem in its pure state the reader will gain an insight which it is hoped will help him when he comes to apply it to his field of work. The actual process of setting up the equations and of interpreting the solution is one which more properly belongs to that field, and in any case is a problem of a different nature altogether. So we need not concern ourselves with it here and are able to concentrate on the mathematical aspect of the situation. The most important tools for handling linear equations are vectors and matrices, and their basic properties are developed in separate chapters. The method by which the nature of the solution is described is one which leads immediately to a solution in practical cases, and it is a method frequently adopted when solving problems by mechanical or electronic computers.

**2.2. Review by: Editors.**

*Mathematical Reviews*MR0095190

**(20 #1696)**.

The exposition has been limited to basic notions, explained as fully as possible, with detailed proofs.

**2.3. Review by: Reuben Louis Goodstein.**

*The Mathematical Gazette*

**43**(344) (1959), 140-141.

... I have left to last the volume on Linear Equations, to be able to end on a note of unqualified praise. This is a piece of presentation and exposition of the very highest order, which should be compulsory reading for every student in the first year of a mathematics degree course.

**3. Solid geometry (1961), by Paul M Cohn.**

**3.1. From the Preface.**

Many problems in science and engineering require some knowledge of three-dimensional geometry, and the beat way to acquire such knowledge today is by using vectors and matrices. This method has the advantage that the geometry can then be used to illustrate the algebraic concepts, so that the student can consolidate and extend his knowledge of vector and matrix-theory. The object of this book is to discuss lines, planes, spheres and central quadrics in this way. There is also a chapter on coordinate transformations which are essential to a study of quadrics and which provide a good application of matrix-theory.

The definition and elementary properties of matrices which are needed here may be obtained from the author's

*Linear Equations*(also in this series) or any book on linear algebra. On the other hand, an account of vectors from the geometric point of view has been included; this is to complement the algebraic treatment in

*Linear Equations*and also to make the earlier chapters independent of that text.

It is again a pleasant duty to thank Dr W Ledermann for his helpful suggestions.

**4. Universal algebra (1965), by Paul M Cohn.**

**4.1. From the Preface.**

Universal algebra is the study of features common to familiar algebraic systems such as groups, rings, lattices, etc. Such a study places the algebraic not ions in their proper setting; it often reveals connexions between seemingly different concepts and helps to systematise one's thoughts. The actual ideas involved are quite simple and follow as natural generalisations from a few special instances. However, one must bear in mind that this approach does not usually solve the whole problem for us, but only tidies up a mass of rather trivial detail, allowing us to concentrate our powers on the hard core of the problem.

The object of this book is to provide a simple account of the basic results of universal algebra. The book is not intended to be exhaustive, or even to achieve maximum generality in places where this would have meant a loss of clarity. Enough background has been included to make the text suitable for beginning graduate students who have some knowledge of groups and rings, and, in the case of Chapter V, of the basic notions of topology. Only the final section of the book requires a somewhat greater acquaintance with representation theory.

The discussion centres on the notion of an algebraic structure, defined roughly as a set with a number of finitary operations. The fact that the operations are finitary may be regarded as characteristic of algebra, and its consequences are traced out in Chapter II. Those consequences, even more basic. that are independent of finitarity are treated separately in Chapter I. This chapter also provides the necessary background in set theory, as seen through the eyes of an algebraist.

One of the main tools for the study of general algebras is the notion of

*free algebra*. It is of particular importance for classes like groups and rings which are defined entirely by laws - i.e., varieties of algebras - and this has perhaps tended to obscure the fact that free algebras exist in many classes of algebras which are not varieties. To emphasise the distinction, free algebras are developed as far as possible without reference to varieties in Chapter III, while properties peculiar to varieties are treated separately in Chapter IV.

These two chapters present the only contact the book makes with homological algebra, and a word should perhaps be said about the connexion. The central part of homological algebra is the theory of abelian categories; this is highly developed, but is too restrictive for our purpose and does not concern us here. The general theory of categories. though at an earlier stage of development, has by now enough tools at its disposal to yield the main theorems on the existence of free algebras, but in an account devoted exclusively to algebra these results are much more easily proved directly; in particular the hypotheses under which the theorems a re obtained here are usually easier to verify (in the case of algebras) than the corresponding hypotheses found in general category theory. For this reason we have borrowed little beyond the bare definitions of category and functor. These of course are indispensable in any satisfactory account of free algebras, and they allow us to state our results concisely without taking us too far from our central topic.

The notion of an algebraic structure as formulated in Chapter II is too narrow even in many algebraic contexts and has to be replaced by that of a

*relational structure*, i.e., a set with a number of finitary relations defined on it. Besides algebraic structures themselves, this also includes structures with operations that are many-valued or not everywhere defined. In recent years, relational structures satisfying a given system of axioms, or

*models*, have been the subject of intensive study and many results of remarkable power and beauty have been obtained. With the apparatus of universal algebra all set up, this seemed an excellent opportunity for giving at least a brief introduction to the subject, and this forms the content of Chapters V-VI.

The final chapter on applications is not in any way intended to be systematic; the aim was to include results which could be established by using the earlier chapters and which in turn illuminate the general theory. and which, moreover, are either important in another context (such as the development of the natural numbers in Vll.1 or the representation theory of Lie and Jordan algebras in VII.5-7), or interesting in their own right (e.g., Malcev's embedding theorem for semigroups).

Although the beginnings of our subject can be found in the last century (A N Whitehead's treatise with the same title appeared in 1898), universal algebra as understood today only goes back to the 1930's, when it emerged as a natural development of the abstract approach to algebra initiated by Emmy Noether. As with other fields, there is now a large and still growing annual output of papers on universal algebra, but a curiously large portion of the subject is still only passed on by oral tradition. The author was fortunate to make acquaintance with this tradition in a series of most lucid and stimulating lectures by Professor Philip Hall in Cambridge 1947-1951, which have exercised a much greater influence on this book than the occasional reference may suggest. In other references an easily accessible work has often been cited in preference to the original source, and no attempt has been made to include remarks of an historical character; although such an attempt would certainly have been well worth while, it would have delayed publication unduly. For the same reason the bibliography contains, apart from papers bearing directly on the text, only a selection of writings on universal algebra. This was all the more feasible since a very full bibliography is available in

*Mathematical Reviews*: besides, a comprehensive bibliography on universal algebra is available in G Grätzer.

The book is based on a course of lectures which I gave at Yale University in 1961-1962. I am grateful to the audience there for having been such good listeners, and to the many friends who have performed the same office since then.

**4.2. Review by: Barron Brainerd.**

*Amer. Math. Monthly*

**74**(7) (1967), 878-880.

The book under review is, to the reviewer's knowledge, the first book since Whitehead's which is devoted entirely to the subject in full generality, and therefore it deserves some attention. It has already been reviewed in the Math. Reviews by J R Isbell and in the J. London Math. Soc. by Graham Higman.

Since Isbell's review covers the content quite adequately, I will not detail a list of the subject matter covered. Instead, I shall concentrate on the relevance to mathematics of the subject as a whole and of the book in particular. In his review, Higman remarks: "Universal Algebra is something everyone ought to know about, though nobody should specialise in it." To a large extent this is also the present reviewer's opinion. To develop a new argument or result in, say, group theory and then to generalise it to a wider class of universal algebras, puts a result of proven utility at the disposal of workers studying algebras of this wider class, and hence is a useful contribution to mathematics. On the other hand, to obtain a result about universal algebras as such, without reference to an application, may result in yet another hollow addition to the already overblown mathematical literature.

The book under review is concerned with results of the former kind, i.e. standard results and arguments given in their broadest generality and results proved in a general setting which have obvious application to well-known problems. The reader with a year of graduate work in algebra will find the book a compendium of familiar arguments and proofs arranged in such a way that it is easy for him to fit them into algebra (and mathematics) as a whole. In general, I would say that the book could be used as a basis for a first or perhaps second year graduate course in algebra, for those interested in the relationship between abstract algebra and the foundations of mathematics.

...

Despite some minor imperfections which can easily be rectified in future editions, this book is an important addition to mathematical literature and should be placed in every college and university mathematics library.

**4.3. Review by: John R Isbell.**

*Mathematical Reviews*MR0175948

**(31 #224)**.

The book is tightly organised. The first 107 pages are, as claimed, suitable for beginning graduate students, comprising Chapter I, Sets and mappings (with Grothendieck set theory, so that nothing shall hinder formation of lattices of all categories of algebras in a universe), and "the central part of the book'', Chapter II, Algebraic structures. ... Chapters III and IV concern mainly the algebraic theory of varieties, but Chapter III (Free algebras) includes, for later use, more general criteria for existence of free algebras and other universal objects and, for exercise, substantial results on solvable word problems. Most of Chapter IV consists of an ample treatment of Birkhoff's theorem and related notions, and an athletic treatment of categories derived by changes of operations (such as Lie from associative algebras). ... Chapter V, presents the elements of model theory up to the compactness theorem with virtually no use of formal logic (no completeness theorem). Ultraproducts do the work. Chapter VI covers the main results relating the structure of an axiomatic class of models (particularly, of algebras) to the form of axioms defining it, and related results... The final chapter, Applications, falls roughly into three parts of increasing length, treating abstract dependence relations, the Malcev conditions for embedding a semigroup in a group and related matters, and the problem of special Jordan algebras with its relatives. No one in the field will wish to be without this book.

**4.4. Review by: Abraham Robinson.**

*The Journal of Symbolic Logic*

**34**(1) (1969), 113-114.

Nowadays, the name "universal algebra" in the narrow sense is applied to the theory of algebraic structures which are given by n-ary operations and which may be restricted axiomatically by specified identities between these operations. While this is, in fact, one of the central topics of the book under review, the author has chosen to interpret the term in a wider sense and, in particular, has included sections which will give the reader at least a taste of category theory on the one hand and of model theory on the other hand. Specialists in these fields may feel that even in dealing with universal algebra in the restricted sense, the author should have placed more emphasis on their particular point of view. However, no good text can go in all directions at once and an author should be accorded the right to select his own approach and material as long as the result fulfils its principal purpose and is readable, useful, and stimulating. In all these respects, the book is remarkably successful.

The opening chapter contains an elementary account of axiomatic set theory and of categories. A detailed treatment of universal algebras, including the isomorphism theorems for algebras and a lattice-theoretical formulation of the Krull-Schmidt theorem is given in the second chapter. Free algebras are treated next and this is followed by a chapter on varieties (equationally definable classes) of algebras. At this point, the introduction of the basic notions of model theory - the language of the lower predicate calculus and the notion of a relational structure - becomes natural since the concept of a variety of algebras does not even comprehend the class of commutative fields. As is entirely appropriate in the present context, no deductive calculus is developed and the compactness theorem is proved by means of ultra-products. As for the choice of particular topics of model theory, the reviewer ventures to suggest, in spite of the principle of tolerance formulated earlier, that the inclusion of more details on complete theories, with applications to algebra, might have been more suitable than some of the topics actually given in the book.

**5. Morita equivalence and duality (1968), by Paul M Cohn.**

**5.1. Review by: Fred E J Linton.**

*Mathematical Reviews*MR0258885

**(41 #3530)**.

These notes provide an introduction to, and an exposition of, the ideas of Morita, Bass, Chase, Schanuel, et al., regarding the question, to be answered in terms of the ground ring, of what module categories are equivalent, respectively dual, to each other. The main goal is to provide a good presentation of the equivalence theory; to this end, the notes of Hyman Bass ["The Morita theorems"] are relied upon fairly heavily. A preliminary goal is to develop enough category theory to get by with; the development is in fact very rapid, though not too sketchy, but a little misleading or garbled in places ... A subsidiary goal is the duality question, which receives an adequately careful treatment and is used to open up the related area of quasi-frobeniosity.

**6. Lectures on algebraic numbers and algebraic functions (1969), by Paul M Cohn.**

**6.1. From the Preface.**

These are the notes of a course I gave at London University in 1966-67. These were typed from a manuscript made after the lectures and I am most grateful to Professor Paulo Ribenboim for getting the notes prepared and printed at this distance , and for removing some of the obscurities in the process . I should also like to thank Mrs E M Wight of the Mathematics Department at Queen's University for the excellent typing job she has done.

**6.2. Review by: Andrew P Ogg.**

*Mathematical Reviews*MR0244255

**(39 #5572)**.

This is a set of lecture notes at about the first-year graduate level. The first chapter covers absolute values, Dedekind domains, and the unit theorem and the finiteness of class number for global fields (number fields or function fields in one variable over finite fields)

*à la*Artin-Whaples. The second chapter, which can be read independently of much of the first chapter, concerns function fields in one variable - the Riemann-Roch theorem, elliptic function fields, Abel's theorem. Thus, despite the title, the notes are mainly about algebraic functions; there is very little about number fields. The notes seem readable and generally well-written.

**7. Free rings and their relations (1971), by Paul M Cohn.**

**7.1. Review by: Leonid A Bokut.**

*Mathematical Reviews*MR0371938

**(51 #8155)**.

This book deals with the results obtained in recent years by the author, G. Bergman and others on free associative algebras and related class of rings, first of all free ideal rings (firs). All the rings are associative with 1. The book consists of 8 chapters, chapter 0 and two appendices. ... On the whole, the book is a notable event in the literature of modern algebra. It completes the formation of the theory of free associative algebras and related classes of rings as an independent domain of ring theory.

**7.2. Review by: David J Fieldhouse.**

*Amer. Math. Monthly*

**80**(5) (1973), 573.

In a basic paper in 1964 P M Cohn defined the concept of a right fir or free ideal ring: a ring (with I) in which all right ideals are free and of unique rank. The ensuing years have seen a series of papers, mostly by Cohn and G M Bergman, which have developed this field. It is especially valuable to have these results, some previously unpublished results from Bergman's 1967 Harvard thesis, as well as new results, collected together in a book whose prerequisites are quite minimal (apart from the usual "mathematical maturity" presupposed). Before proceeding to a more detailed account of the book, one might remark that its object is the study of free associative algebras and related rings. The study of such algebras, which is essentially the study of non-commutative polynomial rings over skew fields, will hopefully in the future shed some light on solutions of algebraic equations in non-commuting indeterminants with coefficients in skew fields. This in turn would be the basis of non-commutative algebraic geometry.

**7.3. Review by: A Czerniakiewicz.**

*Bull. Amer. Math. Soc.*

**79**(1973), 873-878.

Free associative algebras, i.e., polynomial rings in noncommuting variables, are not mentioned in most of the standard texts in ring theory; Cohn's book is the first comprehensive treatment of this subject. The book is up-to-date, very well written and essentially self-contained.

The lack of reference to free rings in previous works is understandable; these rings stand somewhat apart from the traditional branches of ring theory. The lack of fmiteness conditions in free algebras (in fact they are automatically infinite dimensional) dissociates them from the classical theory of noncommutative algebras which were always assumed to be finite dimensional vector spaces over the base field. On the other hand, the noncommutativity of free rings separates them from the other main branch of ring theory: algebraic number fields and their generalisations.

Recently, a new way of approaching problems in algebraic geometry has provided a geometric insight into many notions of ring theory. But the commutativity restrictions cannot be easily circumvented. Non- commutative algebraic geometry is almost nonexistent, and before such a theory could be done one would have to develop a theory of algebraic equations in noncommuting indeterminates. At present, not enough is known about the structure of free rings, and we are far from being able to handle some of the most elementary problems that occur in this field. Most of the questions one asks are simple translations from the commutative case and from the theory of free groups. Unfortunately, the methods cannot be successfully adapted in most cases, and the translated "theorems" are not even true.

The systematic development of the subject has been done mainly by P M Cohn during the last decade. Many of the theorems in this book are taken directly from the author's papers and from G M Bergman's thesis (Harvard 1967), in which some outstanding open problems were solved and existing results were simplified and generalised. Though this book deals with a very specialised subject within ring theory, the author has been careful, at every stage, to establish the connection with the traditional branches by showing how many of the standard notions can be generalised.

**8. Skew field constructions (1973), by Paul M Cohn.**

**8.1. From the Publisher.**

These notes describe methods of constructing skew fields, in particular the coproduct co-construction discovered by the author, and trace out some of the consequences using the powerful coproduct theorems of G M Bergman, which are proved here.

**8.2. Review by: Sigurd Elliger.**

*Mathematical Reviews*MR0382340

**(52 #3225)**.

In this 4-week short lecture course, the author deals with four methods of embedding rings in fields. ... Without claiming new results, this presentation has all the features of a good lecture course and is well suited as an introductory reading.

**9. Algebra Volume 1 (1974), by Paul M Cohn.**

**9.1. From the Preface.**

Although algebra has a long history. it has undergone some quite striking changes in the past few decades. Not least among these is the way the subject has entered into the development of other branches of mathematics, over and above its new applications elsewhere. Its changing role is reflected in the importance of algebra in the curricula, as well as in the many excellent textbooks that now exist. Most of these arc designed for undergraduates at North American universities and are either (a) a very broad introduction to linear algebra, with a little groups and rings, for general students taking mathematics, or (b) a course for graduates, or junior-senior students majoring in mathematics, who have already taken a course of type (a). The pattern in Britain is a little different: the honours student specialising in mathematics takes algebra for two or three years (depending on his ultimate interests) and his need is for a textbook which combines (a) and (b) above and is somewhere between them in level. The object of the present work is to provide such a book: the present first volume includes most of the algebra taught in the first two years to undergraduates at British universities: this will be followed by a second volume covering the third year (and some graduate) topics.

The actual prerequisites arc quite small: students coming to this book will normally have met calculus and some analytic geometry, complex numbers and a little elementary algebra (binomial theorem. quadratic equations, etc.). In any case, some of the topics will be familiar ideas in a new form. There is no doubt that the chief difficulty for the student is the abstractness of the subject so some pains have been taken to motivate the ideas introduced. Connexions between different parts of the subject have been stressed and, on occasion, important applications are briefly discussed. There are numerous exercises, ranging from routine problems to further developments or alternative proofs of results in the text. Some of the harder ones arc marked by an asterisk.

The central ideas are group and ring: once the example of the integers (and the integers mod m) has been described in Ch. 2, the basic properties of groups and rings arc developed in Chapters 3, 9 and Chapters 6, 10 respectively. The other main topic is linear algebra. Ch.4 describes vector spaces (without mention of a metric or determinants) and this is followed by a brief account of methods of solving linear equations (Ch.5). Determinants are introduced in Ch.7; although they have not been needed so far, they provide an important invariant and have regained in recent years some of their theoretical importance. Ch.8 deals with metric questions (quadratic forms, Euclidean spaces) and Ch.11 discusses the various normal forms for matrices. In the first chapter the all-important ideas of set and mapping are briefly described, as well as some notions from formal logic. Although not explicitly used in what follows, the latter has had an important influence and some degree of awareness may preserve the reader from pitfalls. There is no separate chapter on categories in this first volume, but the basic terms are introduced to help systematise the results obtained.

It is clearly impossible to be in any sense comprehensive; even if it were possible, this would not be the place. Any selection of material must necessarily be governed by personal taste; my aim has been, if possible. to sustain the reader's interest, while introducing him to the ideas which are important and useful in present-day mathematics. A more detailed idea of the contents may be gleaned from the table of contents and the index.

A book written at this level clearly owes a great deal to other people besides the author (who merely acts as collector, or rather, selector of the material). From the many inspiring lectures I have heard, let me single out the most recent: a delightful series by Paul Halmos in St Andrews, at the time I was collecting exercises. I am grateful to the Senate of London University for allowing me to draw on past examination papers. The manuscript was read by Walter Ledermann, and by Warren J Dicks, who also read proofs; my thanks go to both of them for their helpful comments.

**9.2. Review by: John D P Meldrum.**

*The Mathematical Gazette*

**59**(407) (1975), 53.

Professor Cohn has written this book with the object of providing a textbook which would cover all the algebra taught to an undergraduate doing honours mathematics at a British university. In this he has succeeded, as the topics covered include a reasonable course for an undergraduate. Chapter 1 starts with sets and mappings, including equivalence relations. In Chapter 2, the integers and rational numbers are examined in some detail, and the integers mod m and finite fields are introduced. This provides a source of examples for the later chapters. An introduction to groups comes next, with the emphasis on permutations and symmetry. The next two chapters cover vector spaces and linear mappings and linear equations. Chapter 6 introduces rings and fields, with some emphasis on polynomials. All rings have an identity. The next two chapters are on determinants and quadratic forms. Some more group theory comes in Chapter 9 with the Jordan-Holder theorem, the Sylow theorems and an introduction to free groups. Chapter 10 on rings and modules has as its main result the decomposition of modules over a principal ideal domain. The book ends with a chapter on normal forms for matrices, applying the methods developed in the previous chapter to obtain the Jordan canonical form.

As a textbook for undergraduates, this book has some shortcomings. It is a bit too sophisticated for the average honours undergraduate. This is probably due in part to the need for keeping the book down to a reasonable length. Some of the points that need to be driven home to most undergraduates are dealt with fairly quickly. An example is the proof of the law of exponents in a monoid. Some of the proofs are very neat an advanced and may be difficult for some of the poorer students to grasp. Category theory makes an appearance. While this provides very useful unifying concepts, I would expect it to confuse many undergraduates. In short, this book would probably prove to be very good for the good student, but a bit too advanced for the average undergraduate.

From the point of view of the lecturer, this is a very good book. It provides a fresh approach to m any topics, and unifies m any strands of the theory. There are m an examples at all levels of difficulty and the shortcomings mentioned above can easily be dealt with by the lecturer when giving a course.

...

It is a book which I enjoyed reading, and which demonstrates the wide mastery Professor Cohn has of his subject. All algebraists will find it a useful addition to their library. It is well produced with few misprints, and by present day standards not unduly expensive. I look forward to the second volume which we are promised.

**9.3. Review by: Martin H Pearl.**

*Mathematical Reviews*MR0360046

**(50 #12496)**.

This is the first volume of a proposed two volume set that is intended as a text for honours students in Great Britain. Although very similar in style and format to the many American books intended for seniors or first year graduate students there is a difference in emphasis. Whereas the American curriculum generally contains separate abstract algebra and linear algebra courses, the two subjects are integrated in the present text. ... It is planned that the second volume will deal with cardinal numbers, lattices, categories, Galois theory, valuations, multilinear algebra and ideal theory.

**10. Skew field constructions (1977), by Paul M Cohn.**

**10.1. From the Preface.**

The history of skew fields begins with quaternions, whose discovery (in 1843) W R Hamilton regarded as the climax of his far from ordinary career. But for a coherent theory one has to wait for the development of linear associative algebras; in fact it was not until the 1930's that a really comprehensive treatment of skew fields (by Hasse, Brauer, Noether and Albert) appeared. It is an essential limitation of this theory that only skew fields finite-dimensional over their centres are considered.

Although general skew fields have made an occasional appearance in the literature, especially in connexion with the foundations of geometry, very little of their properties was known until recently, and even particular examples were not easy to come by. The first well known case is the field of skew power series used by Hilbert in 1898 to illustrate the fact that a non-archimedean ordered field need not be commutative. There are isolated papers in the 1930's, 1940's and 1960's (Moufang, Malcev, B H Neumann, Armitsur and the author) showing that the free algebra can be embedded in a skew field, but the development of the subject is hampered by the fact that one has no operation that can be performed on skew fields (over a given ground field) and again produces a skew field. In the commutative case one has the tensor product, which leads to a ring, from which fields can then be obtained as homomorphic images. The corresponding object in the general case is the free product and in the late 50's the author tried to prove that this could be embedded in a skew field. This led to the development of firs (= free ideal rings); it could be shown (1963) that any free product of skew fields is a fir, but it was not until 1971 that the original aim was achieved, by proving that every fir is embeddable in a skew field, and in fact has a universal field of fractions. Combining these results, one finds that any free product or 'coproduct' of fields has a universal field of fractions, or a field coproduct, as we shall call it. It is this result which forms the starting point for these lecture notes.

As the above description shows, the proof of the existence of field coproducts falls into two parts, showing (a) that the coproduct of skew fields is a fir, and (b) that every fir has a universal field of fractions. Of these, (b) was proved and discussed very fully in the author's 'Free Rings'; for this reason the account given below (in Ch.4) leaves out some of the longer proofs. Thus the chapter can serve as a reminder of, replacement for or introduction to 'Free Rings', as the case may be. The result (a) was first proved by the author in 1963, but in 1974 Bergman gave a very far-reaching generalisation, and it is Bergman's results which we present here, in Ch.5, in a simplified form, for which I am indebted to W Dicks. Ch.5 also contains some applications, including a simple example of a skew field extension with different left and right degrees (Artin's problem). The construction of Ch.4 allows us to give a general discussion of skew field extensions in terms of presentations (Ch.6), including the notion of algebraic closure and the word problem: we shall see that the word problem for free fields can be solved (at least in a relative sense) and also meet a simple example, due to Macintyre, of a skew field with unsolvable word problem.

One of the central problems of skew field theory is the solving of equations, and this makes it important to have a good notion of specialisation. Unfortunately specialisation in skew fields lacks many of the good features we are accustomed to, and very little is known about it so far. However a recent study by Bergman (in which he uses his work with Small on polynomial identities) has led to some remarkable and surprisingly detailed results on the structure of specialisations, and these form the content of Ch.7; I am grateful to C M Bergman for allowing me to include this material before publication.

In Ch.8 we come to the actual solving of equations; the results so far are quite meagre, but it seemed worthwhile at least to set out the problems in what may be a more accessible form. The main results include the similarity reduction of a matrix over a skew field, and the reduction of the main problem (solving equations) to an eigenvalue problem. Some of these results have not appeared in print before, but even for those that have, it was usually possible to give a simpler presentation here, in the framework of a general development of the subject.

The first three chapters deal with matters which are better known (though not all have found their way into text-books yet), partly to show their relation to the rest of the notes but also to make the notes more widely accessible. Apart from Ore's method of skew polynomials (Ch.1) and skew power series (Ch.2) it includes a discussion of extensions of finite degree (Ch.3), in particular Amitsur's theory of cyclic extensions, and an account of Galois theory. No special prerequisites are needed, beyond a standard algebra course.

The bibliography includes (besides the works referred to in the text) a number of papers on skew fields infinite-dimensional over their centres, but does not lay claim to completeness.

As the name indicates, these really are lecture notes, though not for a single set of lectures. For this reason they may lack the polish of a book, but it is hoped that they have not entirely lost the directness of a lecture. The material comes from courses I have given in Manchester and London; some parts follow rather closely lectures given at Tulane University (1971), the University of Alberta (1972), Carleton University (1973), Tübingen (1974), Mons (1974), Haifa Technion (1975), Utrecht (1975) and Ghent (1976). It is a pleasure to acknowledge the hospitality of these institutions, and the stimulating effect of such critical audiences.

**10.2. Review by: L A Koifman.**

*Mathematical Reviews*MR0463237

**(57 #3190)**.

These notes present an excellent account of skew field theory, including recent advances in this topic. The essential feature is that the author considers arbitrary skew fields rather than finite-dimensional ones over their centres as in the classical theory due to Hasse, Brauer, Noether and Albert. This requires methods quite different from the classical ones. The notes are essentially based on methods and results of the author's monograph [Free rings and their relations] and can be regarded as its natural continuation. ... The book is very well written {perhaps with one exception in Chapter 7, where the reader must guess at the definition of a rational identity and a rational function with coefficients in a skew field}. Throughout the notes, the author succeeds not only in giving a coherent account of the subject matter at hand but also in bringing the reader to the frontiers of the theory and in stimulating him to do further research in the indicated spectrum of problems.

**10.3. Review by: George M Bergman.**

*Bull. Amer. Math. Soc.*

**1**(1979), 414-420.

A skew field, or division ring, is an associative ring with 1 in which every nonzero element is invertible. Why do we study such objects?

A natural answer is that they generalise fields, which are so important in commutative ring theory. But this analogy is not as strong as it seems. Skew fields generalise fields as characterised by the property that every nonzero element is invertible. But commutative fields have a number of other important characterisations. They are, for instance, the simple commutative rings. A consequence is that every nonzero commutative ring $R$ has homomorphisms into fields, and the class of such homomorphisms (up to an obvious equivalence) gives a basic framework for studying $R$: It is (with a few extra trimmings) the "prime spectrum" $Spec R$ of algebraic geometry. ...

The major breakthrough in the study of the noncommutative situation was P M Cohn's discovery in 1970 that a skew field $D$ generated by a homomorphic image of a ring $R$, though not generally determined by the set of elements of R which go to zero in $D$ (we have noted that $D$ may not be unique when this set is {0}) is determined, up to natural isomorphism, by the set of square matrices (of all sizes) over $R$ which become singular over $D$! The key idea, which was borrowed from work on rational noncommuting formal power series by M Schützenberger, M Nivat et al., who in turn borrowed it from the theory of differential equations, is that of transforming a single complicated equation into a system of linear equations in a larger number of variables ...

In general, the book is a useful introduction to large areas of current research in skew fields. It does not emphasise open problems but it should leave the reader prepared to read current articles at the cutting edge of the field.

**11. Algebra Volume 2 (1977), by Paul M Cohn.**

**11.1. From the Publisher.**

Cohn's Algebra is a textbook in two volumes which introduces, and covers with great authority, those essential and important topics in algebra which are likely to be encountered by undergraduate and early postgraduate students of mathematics. The first volume dealt with mainly introductory topics and laid a basis for the more advanced work covered in the second volume.

This, the second volume of the textbook, presents topics which are usually found in an advanced undergraduate or postgraduate course in modern algebra. The material falls roughly into three parts - basic theories, fields and rings.

Many of the topics are taken here further than in other texts. While preserving the character of an introduction, this textbook gives substantial accounts of many different parts of algebra, including such topics as lattice theory, homological algebra, and Noetherian and PI-rings, which are not always included at this level in other books. Even for such standard topics as Galois theory or simple algebras, the book includes some new proofs, as well as new presentations which aim to simplify the exposition.

**11.2. Review by: John D P Meldrum.**

*The Mathematical Gazette*

**62**(420) (1978), 137.

When reviewing the first volume of this work, I said that I looked forward to the second volume. It is a pleasure to be able to say that I have not been disappointed. Professor Cohn has written a very good successor to his earlier volume, and it has been very well produced by Wiley. Indeed the number of misprints which I was able to find is remarkably small.

This book is intended to cover topics taught in the final year of an honours undergraduate course or at postgraduate level. Obviously this has meant a certain amount of selection. One really significant omission is group theory. But as this subject is well covered in other books, and as its inclusion would have meant either an outsize book or the omission of much else of importance, the lack of group theory in this particular volume does not diminish its value. There is a good introduction to set theory and lattices, followed by tensor products and graded algebras and a chapter on homological algebra. The next four chapters are on various aspects of field theory, namely Galois theory, further field theory (mainly infinite extensions), real fields and quadratic forms. The final chapters cover ring theory, starting with valuation theory, then artinian rings commutative rings and noetherian and PI rings.

As would be expected by all those who have read the first part of this work, the treatment is clear, thorough and often throws new light on the subject. There are a number of deep and interesting results proved, for instance the Golod-Shafarevich theorem on graded algebras, Wedderburn's theorem on finite division rings, Sturm's theorem on the number of zeros of a real polynomial, Goldie's theorem on rings of fractions and Razmyslov's theorem on non-constant central polynomials, to name but a few. There are a number of exercises in each chapter. These are mainly to expand or amplify the results given in the text.

A number of advanced courses in algebra could be based on this book. Indeed most courses except those on group theory could use this text. So it will prove of great use to any lecturer in algebra. A postgraduate student or an honours undergraduate with a real interest in algebra would find this book very useful. The reader will have to work hard if he is to make the best use of it, as the work is sometimes very condensed. But this is to be expected at this level. The lecturer will find it invaluable as a comprehensive textbook.

I am sure that the two volumes will soon become a standard reference work in algebra. The coverage is very wide, the treatment elegant and the approach very much up to date. One of the more pleasant aspects is the very reasonable price. The two volumes together come to less than many single volumes trying to cover similar ground. I can recommend this book without reservations.

**11.3. Review by: Leo C A van Leeuwen.**

*Mathematical Reviews*MR0530404

**(58 #26625)**.

... the reviewer would first like to comment on the way in which the concepts are introduced. In many cases this is not the standard way. The author says in his preface that the book was fun to write and that, as he hopes, it will be fun to read. Because of its originality, the reviewer did find the book fun to read. Often a new proof of a well-known theorem is given, e.g., Cartan-Brauer-Hua. At the end of each chapter there are exercises of two kinds, comprising a number of routine examples and some which the author labels as more difficult. There is also advice on how to use the book for a course in algebra. In reviewing the first volume, the reviewer expressed the hope that the second volume would appear soon. This hope has now been fulfilled. Both volumes together comprise a text which, when mastered by a student, will equip him well to study more advanced topics in algebra.

**12. Universal algebra (Second Edition) (1981), by Paul M Cohn.**

**12.1. From the Preface.**

The present book was conceived as an introduction for the user of universal algebra, rather than a handbook for the specialist, but when the first edition appeared in 1965, there were practically no other books entirely devoted to the subject, whether introductory or specialised. Today the specialist in the field is well provided for, but there is still a demand for an introduction to the subject to suit the user, and this seemed to justify a reissue of the book. Naturally some changes have had to be made; in particular, I have corrected all errors that have been brought to my notice. Besides errors, some obscurities in the text have been removed and the references brought up to date.

**12.2. Review by: Walter F Taylor.**

*Mathematical Reviews*MR0620952

**(82j:08001)**.

This is a reissue, after fifteen years, of the author's book of the same title. It contains four new chapters in a 70-page supplement; the original text is unaltered except for a few needed corrections and some updatings. The four new chapters are: Category theory and universal algebra, Model theory and universal algebra, Miscellaneous further results, and Algebra and language theory. ... In the last fifteen years Universal algebra has aspired to, and accomplished, many things, often concerning some very unfamiliar algebras. And so the book under review remains as praiseworthy as ever for its well-written unification of classical algebraic themes, but gives an inadequate picture of what UA is like today. The four new chapters are very readable and informative, ..., but they are almost entirely the product of other mathematical disciplines (category theory, model theory, automaton theory), as the author has taken some care to point out. It is thus unfortunate that the publisher's book jacket makes the entirely misleading claim that these new chapters "reflect the activity in Universal algebra in the time between the two editions''. None of the central activity - e.g., congruence identities, congruence representations - is to be found in this book; the reader will have to seek it elsewhere.

**13. Algebra Volume 1 (Second Edition) (1982), by Paul M Cohn.**

**13.1. From the Preface.**

The present edition offers the first opportunity to make substantial changes to the text. In addition to a complete set of answers to the exercises, these changes are of three kinds. In the first place some additions have been made, dealing with affine spaces, linear programming, a more extensive treatment of duality and a second, more direct derivation of the Jordan normal form. Secondly, in a number of places concepts have been introduced which should form part of the algebraist's education, mainly in group theory which is not taken further in Vol. 2. And thirdly, some obscurities have been removed, proofs expanded, worked examples added and, of course, some further errors corrected.

l am indebted to many colleagues for their helpful criticism of the book, in particular, G M Bergman has sent extensive observations after using the book as a text, J C Fernau and W Stephenson have made useful suggestions on 5.3 and W Dicks helped with proof-reading. I am also grateful to many students (including the dedicatees) who by their queries helped me to spot and eradicate obscurities in the text. Finally I would like to thank the publishers for their willingness to undertake this reissue.

**14. Free rings and their relations (Second Edition) (1985), by Paul M Cohn.**

**14.1. From the Preface.**

The present edition has been completely reset, and this has given me an opportunity to make substantial changes. In many respects it is now possible to see more clearly how the subject ought to be developed; the resulting changes, while in themselves small, have affected nearly every page and have, I hope, helped to make the book easier to read. At the same time a number of new developments have been included, particularly: (i) Sylvester domains, (ii) localisation

*à la*Gerasimov and Malcolmson, (iii) automorphisms of free algebras, (iv) normal forms of matrices over free algebras and (v) a closer study of the universal field of fractions of a semifir. Some older topics, which now have more accessible proofs, have been included, such as (vi) the computation of the dependence number, leading to a simple construction of n-firs with prescribed properties, (vii) the specialisation lemma in a new and more general form and (viii) the Malcev-Neumann construction. To give one example, the embeddability of a free algebra in a field, for which there were two proofs in the old edition, has five proofs in the present one. There have also been many smaller innovations, both in the forms of the results and in the proofs. Of the 86 open problems of the first edition, 28 have been solved; most of the solutions are included here, as well as some new problems.

These changes have inevitably caused the volume to grow, but I have been conscious of the need to keep the size within bounds. However, I did not feel justified in leaving out topics from the first edition (except for some better-known background results, for which a reference is readily available), in order to avoid a situation where the first edition continues as a reference for the parts omitted from the second. As in the previous edition, the bibliography is selective but aims to include most of the relevant recent papers; this has meant that some papers previously included have now been omitted, and the total has not quite doubled in size.

It goes without saying that all known errors have been corrected; in this task I have been helped by many correspondents, to whom I would like to express my appreciation. My thanks also go to L A Bokut', who was responsible for the Russian translation and who provided me with a list of errors as well as further references. I have continued to benefit from George M Bergman's comments on the work, as well as his very substantial help with recent papers of mine which are used in the text. From October 1982 to June 1984 the Bedford College Ring Theory Study Group went through the manuscript on which this edition is based, and I am grateful to the participants for their help and persistence. I owe a special debt to Warren Dicks who played a major role in the Study Group; in particular, he presented his own account of automorphisms and invariants of free algebras, and his notes have (with little change) formed the basis of sections 6.8-11. He also read and commented on substantial parts of the manuscript. This often saved me from error and helped me to see the wood among the trees. I have been particularly conscious of his help with homological proofs, which are briefer and clearer as a result. The manuscript has also been read by Mark L Roberts and both he and Mark Hedges have read the proofs and in the process caught a number of errors; I am grateful for their help. Finally I should like to thank the staff of Academic Press for the very efficient way they have coped with the publication of this volume.

**14.2. Review by: Leonid G Makar-Limanov.**

*Mathematical Reviews*MR0800091

**(87e:16006)**.

This is the second edition of a book proven to be rather important in developing the subject of free (associative) algebras. Its importance is not only as a source for learning and reference but also as a collection of attractive open questions. The first edition was reviewed in great detail [1971], so we shall mention only the major additions. They are: a theory of Sylvester domains; automorphisms of free algebras (including a description of the group of automorphisms of a free algebra with two generators); a far more substantial theory of the universal field of fractions of a semifir; and localisation in the style of Gerasimov and Malcolmson. Apart from that, numerous improvements have been made throughout all parts of the book. The reviewer highly recommends this book to anyone interested in ring theory, especially to graduate students.

**14.3. Review by: Jacques Lewin.**

*Bull. Amer. Math. Soc.*

**21**(1989), 139-142.

The fundamental theorem of combinatorial group theory is that a subgroup of a free group is free. In contrast, subalgebras of free (associative) algebras are not well understood, and at the moment defy classification. This is not too surprising: in going from group theory to ring theory the translation of "subgroup" is (one-sided) "ideal." Thus the correct, and fundamental, theorem is that one-sided ideals of free algebras are free submodules. This is a consequence of a "weak algorithm" ... [which] can be investigated in any graded ring, and hence in any filtered ring, and the notion can be refined to that of n-term weak algorithm (bound the number of summand in the definition). It is the discovery and investigation of the consequences of weak algorithms which marks the beginning of the subject treated in this monograph. That the algorithms appear in Chapter 2 reflects the fact that the above mentioned consequence is of more fundamental importance. ... This text is an invaluable tool for the researcher and the diligent reader will find it quite rewarding.

**15. Algebra Volume 2 (Second Edition) (1989), by Paul M Cohn.**

**15.1. Review by: Haya Freedman.**

*The Mathematical Gazette*

**75**(471) (1991), 122.

The present volume is a revised edition of

*Algebra*, volume 2, published in 1977. It has been rewritten to include topics for third year undergraduate courses, while part of the more advanced material has been reserved for volume 3 intended for postgraduate students. The main omissions are: chapters 10 and 12, the best part of chapters 6 and 8 and about half of each of chapters 1, 3, 4, and 7 of the old volume 2. The material omitted includes the more advanced parts of field and ring theory, topics in homological algebra and Peano's axioms.

The main additions to the present volume are: three new chapters, a few new sections and some important theorems, such as Hensel's lemma and the continuity criterion for $p$-adic functions. In addition some proofs have been simplified and further examples and illustrations provided.

...

The material is well chosen (but it would have been great fun, for example, to find a section on the connection between codes and latin squares). The proofs are usually clear, elegant and some of them are very original. However, occasionally one finds that self evident points are elaborated whereas some more difficult points are glossed over quickly. The only other criticism is the omission of skeleton solutions to the many exercises. However, these are relatively minor points. On the whole this is an important textbook which has been further perfected in this new edition.

**15.2. Review by: Boris M Schein.**

*Mathematical Reviews*MR1006872

**(91b:00001)**.

The first editions of Volumes 1 and 2 have been reviewed [London, 1970; 1977]. The second edition of Volume 1 has also been reviewed [1982]. This is the first part of a completely rewritten and expanded version of Volume 2. The resulting increase in material made it necessary to divide that volume into the present volume and the forthcoming Volume 3. The present volume includes four chapters from the old Volume 2 and about half of another four chapters, and the rest of that first edition is represented by smaller proportions. Even where the organisation has remained the same, there have been many changes in detail. For example, Hensel's lemma, Ramsey's theorem and the continuity criterion for $p$-adic functions have been added. There are three new chapters. A chapter on representation theory replaces a short sketch from the first edition. Another chapter introduces block codes, and there is a chapter on the algebraic language theory, variable-length codes, automata, and power series rings. On the whole, the book is much easier to read, proofs in many places were expanded and simplified, and many examples and illustrations have been added.

**16. Algebra Volume 3 (Second Edition) (1991), by Paul M Cohn.**

**16.1. Review by: Boris M Schein.**

*Mathematical Reviews*MR1098018

**(92c:00001)**.

Volume 2 of the first edition of this book now becomes Volumes 2 [1989] and 3. Volume 3 contains parts of Chapters 1, 3, 4, 6, 7, 8, 10, and 12 from the first edition and new material. Here are some of the main changes. The material on graded algebras is expanded to bring out the role of the universal derivation bimodule, and it contains more on Hilbert series. A discussion of symplectic and orthogonal groups has been added. The chapter on group theory now includes Hall subgroups, free groups, commutators, and linear groups. The chapter on algebras deals with the Krull-Schmidt theorem, projective covers, Morita equivalence and related matters, stopping short of the representation theory of algebras. There is more detail on various radicals of general rings. The final chapter on skew fields contains simplified proofs of results on the Dieudonné determinant and the existence of "free fields'', as well as valuations on skew fields and right dimensions (Artin's problem). There are numerous exercises of various degrees of difficulty as well as brief historical remarks.

**16.2. Review by: Haya Freedman**

*The Mathematical Gazette*

**76**(477) (1992), 426-427.

Algebra: volume 2 was first published in 1977. Since then Professor Cohn has revised and expanded it into two new volumes: volume 2 which was published in 1989 and the present volume 3. The work, aimed at graduate students, contains, in addition to new material, the more sophisticated parts from the old volume 2. The first edition has been very well received and extensively reviewed. ... some proofs are simplified, the selection of exercises modified, remarks and short notes expanded and updated and much of the material reorganised. All this greatly enhances the work and results in a more cohesive structure. There is clearly a bias towards non-commutative algebra, but this does not deter from the value of the book. On the contrary, it gives it a unique flavour and reflects the author's enthusiasm for non-commutative rings and fields.

**17. Algebraic numbers and algebraic functions (1991), by Paul M Cohn.**

**17.1. From the Publisher.**

This book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.

**17.2. From the Preface.**

One of the most interesting and central areas of mathematics is the theory of algebraic functions - algebra, analysis and geometry meet here and interact in a significant way. Some years ago I gave a Course on this topic in the University of London, using Artin's approach via valuations, which allowed one to treat algebraic numbers and functions in parallel. At the invitation of my friend Paulo Ribenboim I prepared a set of lecture notes which was issued by Queen's University, Kingston, Ontario, but I had always felt that they might be deserving of a wider audience.

Ideally one would first develop the algebraic, analytic and geometric background and then pursue the theme along these three paths of its development. This would have resulted in a massive and not very readable tome. Instead I decided to assume the necessary background from algebra and complex analysis and leave out the geometric aspects, except for the occasional aside, but to give a fairly full exposition of the necessary valuation theory. This allowed the text to be kept to a reasonable size.

Chapter 1 is an account of valuation theory, including all that is needed later, and it could be read independently as a concise introduction to a powerful method of studying general fields. Chapter 2 describes the behaviour under extensions and shows how Dedekind domains can be characterised as rings of integers for a family of valuations with the strong approximation property; this makes the passage to extensions particularly transparent. These methods are then put to use in Chapter 3 to characterise global fields by means of the product formula, to classify the global fields and to prove two basic results of algebraic number theory in this context: the unit theorem and the finiteness of the class number.

Chapter 4, the longest in the book, treats algebraic function fields of one variable. The main aim is the description of the group of divisor classes of degree zero as a particular 2g-dimensional compact abelian group, the Jacobian variety. This is the Abel-Jacobi theorem; the methods used are as far as possible algebraic, although the function-theoretic interpretation is borne in mind. On the way automorphisms of function fields are discussed and the special case of elliptic function fields is developed in a little more detail. A final brief chapter examines the case of valuations on fields of two variables; in a sense this is a continuation of Chapter 1 which it is hoped will illuminate the development of Chapter 4. The main exposition follows the notes, but the telegraphese style has been expanded and clarified where necessary, and there have been several substantial additions, especially in Chapter 4.

The prerequisites are quite small: an undergraduate course in algebra and in complex function theory should suffice; references are generally given for any results needed. There are a number of exercises containing examples and

further developments.

In writing a book of this kind one inevitably has a heavy debt to others. The first three chapters were much influenced by the writings of E Artin quoted in the bibliography. A course on number theory by J Dieudonné (which I attended at an impressionable age) has helped me greatly in Chapter 2. The treatment of valuations on function fields of two variables (Chapter 5) is based on a paper by Zariski, while Chapter 4 owes much to several works in the bibliography, particularly those by Hensel and Landsberg, Chebotarev and by Eichler. I should like to thank Paulo Ribenboim for bringing out the earlier set of notes and a number of friends and colleagues for their advice and help, particularly David Eisenbud, Frank E A Johnson and Mark L Roberts. The latter has also helped with the proof reading.

**17.3. Review by: Boris Datskovsky.**

*Mathematical Reviews*MR1140705

**(93e:11001)**.

The purpose of this book is to provide a parallel treatment of the theories of algebraic numbers and algebraic functions. The basic tool for studying both is Artin's theory of valuations, a fairly comprehensive account of which is included in Chapters 1 and 2. In Chapter 3 applications to number theory are pursued as far as the Dirichlet unit theorem and the finiteness of the class number. Chapter 4 contains perhaps the best part of the book: a treatment of the theory of algebraic functions in one variable from scratch, via the Riemann-Roch theorem, and up to the Abel-Jacobi theorem. The concluding chapter looks at valuations of 2-dimensional function fields following a 1939 paper of Zariski. This book contains a number of good exercises and is suitable for use in an upper level undergraduate or a beginning graduate course. However, I personally found the early chapters dry and unmotivating. The real fun begins in Chapters 3 and 4. I feel that in order to appreciate this book fully one ought to be already acquainted either with algebraic number fields or algebraic function fields and their valuations.

**18. Elements of linear algebra (1994), by Paul M Cohn.**

**18.1. From the Publisher.**

This volume presents a thorough discussion of systems of linear equations and their solutions. Vectors and matrices are introduced as required and an account of determinants is given. Great emphasis has been placed on keeping the presentation as simple as possible, with many illustrative examples. While all mathematical assertions are proved, the student is led to view the mathematical content intuitively, as an aid to understanding. The text treats the coordinate geometry of lines, planes and quadrics, provides a natural application for linear algebra and at the same time furnished a geometrical interpretation to illustrate the algebraic concepts.

**18.2. From the Preface.**

In 1958 I wrote a brief account of

*Linear Equations*, which was followed by an equally short book on

*Solid Geometry*. Both books have been surprisingly popular, so in making the first major revision in over 30 years I have endeavoured to retain the simple style of the original. It seemed practical to combine both in one book, since they essentially deal with the same subject from different points of view. The revision also provided an opportunity to include some basic topics left out of the earlier versions.

The core of the book is formed by Chapters 2 and 4, where the solution of systems of linear equations is discussed, taking the regular case (systems with a unique solution) in Chapter 2, followed by the general case in Chapter 4. To facilitate the discussion, vectors are introduced in Chapter 1 and matrices in Chapter 3. All this can be done quite concisely and, together with the account of determinants in Chapter 5, constitutes the old Linear Equations. Of course the whole has been revised and clarified, with additional examples where appropriate.

The next two chapters, 6 and 7, together with parts of Chapters 8 and 9, represent Solid Geometry. The short chapter on spheres from that text has been relegated to the exercises, but transformations have been discussed in more detail, and some attention is given to the n-dimensional case (which is of practical importance in statistics, mechanics, etc.). An account of linear mappings between vector spaces has also been included; it follows on naturally and provides a motivation for the description of normal forms in Chapter 8. This concerns mainly the canonical form of matrices under similarity transformations (Jordan normal form], but also the transformation to diagonal form of symmetric matrices by congruence transformations, which may be regarded as a special case of similarity. Some care has been taken to arrange matters so that the more difficult reduction to the Jordan normal form comes at the end of the chapter and so can be omitted without loss of continuity. Throughout the text there are brief historical remarks which situate the subject in a wider context and, it is hoped, add to the reader's interest.

The remaining chapters, 9 and 10, are devoted to applications. Systems of linear equations are used so widely that it was necessary to be very selective; it would have been possible to present amusing examples from many fields, but it seemed more appropriate to confine attention to cases which used methods of mathematical interest. Chapter 9, on algebraic and geometric applications, deals with the simultaneous reduction of two quadratic forms and the resulting classification of quadric surfaces. Section 9.6, on linear programming, describes the basic problem, of solving systems of linear inequalities, and its solution by the simplex algorithm; like many of these later sections, it can only give a taste of the problem, and the interested reader will be able to pursue the topic in the specialised works listed in the Bibliography at the end. Another normal form, the polar form, is described, and the chapter ends with a brief account of the method of least squares, which provides a good illustration of the use of matrices.

Chapter 10 deals with applications to analysis. The algebraic treatment of linear differential equations goes back to the nineteenth century, and in fact some of the algebraic methods, such as the theory of elementary divisors, were originally developed in this context. This theory goes well beyond the framework of the present book, where we only treat the two basic applications - the use of the Jordan normal form to solve linear differential equations with constant coefficients. with illustrations from economics, and the application of the simultaneous reduction of quadratic forms to find the normal modes of vibration of a mechanical system. Much of the theory has an analogue for linear difference equations, which is described and illustrated by the Fibonacci sequence. A more recent development is the use of algebraic methods in the calculus of functions of several variables, which in essence is the recognition of the derivative as a linear mapping of vector spaces. A brief outline explains the role of Jacobians, Hessians and the use of the Morse lemma in the study of critical points. The contraction principle of functional analysis is illustrated by an iterative method to invert a matrix.

The book is not primarily aimed at mathematics students, but addresses itself to users of mathematics, who would follow up its study with a book devoted to their special field. To quote from the original preface to

*Linear Equations*,

the book is not merely intended as a reference book of useful formulae. Rather it is addressed to the student who wants to gain an understanding of the subject. With this end in mind, the exposition has been limited to basic notions, explained as fully as possible, with detailed proofs. Some of the proofs may be omitted at first, but they do form an integral part of the treatment and, it is hoped, will help the reader to grasp the underlying principles.

To assist the reader, exercises have been included at the end of each chapter, with outline solutions at the end of the book. The reader should bear in mind that a proof is often more easily understood by first working through a particular case.

**19. Skew fields (1995), by Paul M Cohn.**

**19.1. From the Preface.**

When

*Skew Field Constructions*appeared in 1977 in the London Mathematical Society Lecture Note Series, it was very much intended as a provisional text, to be replaced by a more definitive version. In the intervening years there have been some new developments, but most of the progress has been made in the simplification of the proofs of the main results. This has made it possible to include complete proofs in the present version, rather than to have to refer to the author's

*Free Rings and their Relations.*An attempt has also been made to be more comprehensive, but we are without a doubt only at the beginning of the theory of skew fields, and one would hope that this book will offer help and encouragement to the prospective builders of such a theory. The genesis of the theory was described in the original preface (see the extract following this preface); below we briefly outline the subjects covered in the present book.

The first four chapters are to a large extent independent of each other and can be read in any order, referring back as necessary. Ch.1 gives the general definitions and treats the Ore case as well as various necessary conditions for the embedding of rings in skew fields. From results in universal algebra it follows that necessary and sufficient conditions for such an embedding take the form of quasi-identities. Later, in Ch.4, we shall find the explicit form of these quasi-identities, and in Ch.6 we shall see that this set must be infinite. The rest of Ch.1 gives the definition and basic properties of free algebras and free ideal rings, which play a major role later. It also includes some technical results on the association of matrices and it introduces an important technical tool: the matrix reduction functor.

Ch.2 studies skew polynomial rings and the fields formed from them, as well as power series rings and generalisations such as the Malcev-Neumann construction, and the author's results on fields of fractions for a class of filtered rings. Ch.3 is devoted to the Galois theory of skew fields, now almost classical, with applications to (left or) right polynomial equations over skew fields, and special cases of extensions, such as pseudo-linear extensions and cyclic Galois extensions.

Ch.4 is in many ways the central chapter. The process of forming fields of fractions or more generally epic $R$-fields for a ring $R$ is described in terms of the

*singular kernel*, i.e. the set of matrices that become singular over the field. It is shown how any epic $R$-field can be constructed from its singular kernel, while the latter has a simple description as prime matrix ideal. This leads to explicit conditions for the existence of a field of fractions. In particular, the rings with a fully inverting homomorphism to a field are characterised as Sylvester domains and it is shown that every semifir has a universal field of fractions. Of the earlier sections only 1.6 is needed here.

Ch. 5 describes the coproduct construction and the results proved here are basic for much that follows. It is also the most technical chapter and the reader may wish to postpone the details of the proofs in 5.1-3 to a second reading, but he should familiarise himself with the results. They are applied in the rest of the chapter to give the HNN-construction for fields and for rings, to study the effect of adjoining generators and relations, particularly matrix relations, and to construct field extensions with different left and right degrees (Artin's problem). Ch.6 deals with some general questions. There is a study of free fields; here the specialisation lemma is an essential tool. Other topics include the word problem and existentially closed fields.

Ch.7 on rational identities is mainly devoted to Bergman's theory of specialisations between rational meets of X-fields; it is independent of most of the rest and can be read at any stage.

In Ch.8 the rather fragmentary state of knowledge of singularities (which in the general theory take the place of equations in the commutative theory) is surveyed, with an account of the problems to be overcome to launch a form of non-commutative algebraic geometry. Ch.9 deals with valuations and orderings on skew fields from the point of view of the general construction of Ch.4 and it shows for example how to construct valuations and orderings on the free field.

The exercises are intended for practice but serve also to present additional developments in brief form, as well as some open problems. Some historical background is given in the Notes and comments.

The theory of division algebras (finite-dimensional over a field) is very much further advanced than the general theory of skew fields, and a comprehensive account including a full treatment of division algebras would have thrown the whole out of balance and resulted in a very bulky tome. For this reason that topic has largely been left aside; this was all the more reasonable as the subject matter is much more accessible, and no doubt will be even more so with the forthcoming publication of the treatise by Jacobson and Saltman.

Nearly all the material in this volume has been presented to the Ring Theory Study Group at University College London and I am grateful to the members of this group for their patience and help. I would like to thank Mark L Roberts for his comments on early chapters and George M Bergman for his criticism of

*Skew Field Constructions,*which has proved most useful.

**19.2. Review by: Jan van Geel.**

*Mathematical Reviews*MR1349108

**(97d:12003)**.

The work presents the theory of general division rings. Division ring or skew field stands for a ring with unity in which the nonzero elements form a noncommutative group. (Cohn uses the word field as a general term for commutative fields and skew fields.) "General theory'' means that essentially skew fields that are not finite-dimensional over their centre are considered. As a consequence the theory requires methods and techniques which are very different from the one used in the classical theory of Hasse, Brauer, Noether and Albert. Stating that this book is an elaborate version of the author's lecture notes Skew field constructions which appeared in 1977 is an underestimation of the work. As mentioned in the introduction the earlier Skew field constructions was a provisional text meant to be replaced by a more definitive version. The mathematicians interested in this area had to wait patiently 18 years before a comprehensive account of the theory, including an axiomatic foundation of the embedding problem, topological construction methods, Cohn's general embedding theory and G. Bergman's coproduct theorems, appeared. That patience is rewarded with a well-written book, which takes full advantage of the many new developments in the area during these two decades. This progress is also seen in the many simplifications of the proofs, making a self-contained treatment possible. ... The theory of skew fields is still not so familiar as the commutative analogue. The complexity of the problems in the noncommutative setting is one of the reasons for this fact. It is Cohn's merit to provide a coherent treatment of this subject which at the same time leads the reader to a wide range of interesting and important research problems, related to questions in algebra, geometry and logic.

**20. Algebra Vols 1, 2 and 3 (Second paperback edition), (1995), by Paul M Cohn.**

**20.1. Review by Michael C R Butler.**

*The Mathematical Gazette*

**81**(491) (1997), 330-332.

The new volume 1, though some 90 pages longer than its predecessor, still aims to provide undergraduates with a broad introduction to modern algebra, and does so under the same chapter headings as the first edition. Some of the extra pages offer help to readers, partly by easing previously rather condensed arguments, and partly by the inclusion of 37 pages of Solutions to Exercises. Other extra pages introduce important, but closely related additional topics. Thus, the chapter on vector spaces discusses quotients and affine spaces, that on linear equations now includes an introduction to the important topic of linear programming, the revised quadratic forms chapter begins with rather more theory of bilinear forms, the second group theory chapter now introduces primitivity, semi-direct products and nilpotence, and the last chapter includes a second derivation, independent of previous theory, of the Jordan normal form of a matrix. These changes to volume 1, though slight in relation to the whole, certainly enhance its value to both learners and teachers of undergraduate algebra.

The changes to volume 2 are altogether more radical, and are in response to the author's own criticism that 'many quite basic parts of algebra had been omitted or inadequately treated'. Thus the new, far longer, volumes 2 and 3 contain not only expanded and revised treatments of the original material, but also substantial introductions to several topics not previously covered. There are major innovations. The new volume 2 gives coding theory some emphasis, firstly, in a chapter introducing block codes (there is even a glimpse through the Leech lattice of one of Conway's sporadic simple groups), then secondly there is a separate chapter on variable length codes in the context of language theory, automata, and free and power series algebras. Groups are given far more attention than in the first edition, with two entirely new chapters - one in volume 2 on ordinary character theory up to the symmetric groups and the classical theorems of Burnside and Frobenius, and a second more diverse chapter in volume 3. This introduces group extensions and cohomology, more about finite groups (Frattini, Fitting and Hall subgroups, transfer homomorphisms), free groups (including freeness of subgroups, and residual p-finiteness for any prime p), and finally linear groups, including the simplicity of the projective special linear groups. ...

In my opinion, this massively revised and much longer second edition of

*Algebra*is a book of quite outstanding quality and interest, though like the first edition, its pace and level of sophistication probably makes even volume 1 rather too difficult for most undergraduates. More experienced readers will find the book clearly, concisely and elegantly written with its main ideas motivated and illustrated by an exceptionally rich and interesting collection of examples and exercises. Another attractive feature of the book, indeed a feature giving the book a flavour all of its own, is the inclusion of a number of important algebraic topics - for example, lattice theory, universal algebra, algebraic language theory, noncommutative ring theory - often excluded from text books favouring the sort of algebra needed in group theory, algebraic geometry and various representation theories.

**21. Introduction to ring theory (2000), by Paul M Cohn.**

**21.1. From the Publisher.**

Most parts of algebra have undergone great changes and advances in recent years, perhaps none more so than ring theory. In this volume, Paul Cohn provides a clear and structured introduction to the subject.

After a chapter on the definition of rings and modules there are brief accounts of Artinian rings, commutative Noetherian rings and ring constructions, such as the direct product. Tensor product and rings of fractions, followed by a description of free rings. The reader is assumed to have a basic understanding of set theory, group theory and vector spaces. Over two hundred carefully selected exercises are included, most with outline solutions.

**21.2. From the Introduction.**

Most parts of algebra have undergone great changes in the course of the 20th century; probably none more so than the theory of rings. While groups were well established early in the 19th century, in its closing years the term " ring" just meant a "ring of algebraic integers" and the axiomatic foundations were not laid until 1914 (Fraenkel 1914). In the early years of the 20th century Wedderburn proved his theorems about finite-dimensional algebras, which some time later were recognised by Emmy Noether to be true of rings with chain condition; Emil Artin realised that the minimum condition was enough and such rings are now called Artinian. Both E Noether and E Artin lectured on algebra in Göttingen in the 1920s and B L van der Waerden, who had attended these lectures, decided to write a textbook based on them which first came out in 1931 and rapidly became a classic. Significantly enough it was entitled "Moderne Algebra" on its first appearance, but the "Modern" was dropped from the title after 25 years.

Meanwhile in commutative ring theory the properties of rings of algebraic numbers and coordinate rings of algebraic varieties were established by E Noether and others for more general rings: apart from commutativity only the maximum condition was needed and the rings became known as Noetherian rings.

Whereas the trend in mathematics at the beginning of the 20th century was towards axiomatics, the emphasis since the second world war has been on generality, where possible, regarding several subjects from a single point of view. Thus the algebraicisation of topology has led to the development of homological algebra, which in turn has had. a major influence on ring theory.

In another field, the remarkable progress in algebraic geometry by the Italian School has been put on a firm algebraic basis, and this has led to progress in commutative ring theory, culminating in the result which associates with any commutative ring an affine scheme. Thirdly the theory of operator algebras, which itself received an impetus from quantum mechanics, has led to the development of function algebras. More recently the study of "non-commutative geometry" has emphasised the role of non-commutative rings.

To do justice to all these trends would require several volumes and many authors. The present volume merely takes the first steps in such a development, introducing the definition and basic notions of rings and constructions, such as rings of fractions, residue-class rings and matrix rings, to be followed by brief accounts of Artinian rings, commutative Noetherian rings, and a chapter on free rings.

**21.3. Review by Thomas William Hungerford.**

*Mathematical Reviews*MR1732101

**(2001a:16001)**.

At most American universities, this book would be suitable for an introductory course for graduate students or very strong undergraduates who have taken appropriate prerequisite courses. The reader is assumed to be familiar with sets (including equivalence relations), the integers (including the division algorithm, congruence, and congruence classes), groups (including homomorphisms and quotient groups), vector spaces and linear transformations. The presentation is well written, but succinct. A large amount of material is covered in a relatively small space, which could be a problem for some students. Each section is followed by 6-12 nontrivial exercises. Short historical asides on the development of various areas of algebra are a nice touch.

**21.4. Review by: Tony Barnard**

*The Mathematical Gazette*

**85**(503) (2001), 362.

In the subject of algebra, if one is looking for a particular definition, the precise statement of some theorem or the details of a neat proof, a book by Paul Cohn is always a good place to look. This new book on ring theory is no exception. It is a scholarly text containing a large amount of clearly explained material, supplemented by over two hundred pertinent exercises.

Although the introductory sections in principle assume nothing more than an

acquaintance with sets, groups and vector spaces, and include coverage of standard results in commutative ring theory, they are generally written with an eye on more advanced material and also on the non-commutative situation. So whereas it is stated in the preface that the book 'is intended for second and third year students an postgraduates', a more appropriate matching of contents to readers suitability might be as follows.

Chapter 1, on the basics of fields, vector spaces, matrices, modules, categories: suitable for potential first class undergraduate students in their third year.

Chapters 2 and 3, including chain conditions, Wedderbum's structure theorems, group characters, unique factorisation domains, modules over principal ideal domains: suitable for MSci students in their fourth year.

Chapter 4, including tensor products of algebras, hom functors, projective and injective modules: suitable for general algebra MSc and PhD students.

Chapter 5, including rings of fractions of non-commutative rings, skew polynomial rings, free ideal rings: suitable for graduate specialists.

...

For graduates on an MSc algebra programme or doing a PhD in non- commutative rings, this book is indeed a very good 'introduction to ring theory'.

**22. Classic algebra (2002), by Paul M Cohn.**

**22.1. From the Publisher.**

Fundamental to all areas of mathematics, algebra provides the cornerstone for the student's development. The concepts are often intuitive, but some can take years of study to fully absorb. For over twenty years, the author's classic three-volume set, Algebra, has been regarded by many as the most outstanding introductory work available. This work, Classic Algebra, combines a fully updated Volume 1 with the essential topics from Volumes 2 and 3, and provides a self-contained introduction to the subject.

**22.2. Review by: Patrick Quill.**

*The Mathematical Gazette*

**86**(505) (2002), 175.

*Classic algebra*is a single-volume abridged version of

*Algebra*by P M Cohn, published as a three volume set in 1973 and reprinted in 1981. It covers most of the algebra taught in mathematics courses at undergraduate level and some topics from graduate courses.

There are eleven chapters in the book, each divided into sections. Many of these sections (particularly in the second half of the book) are self-contained, and in this way the book serves very well as a reference. Around a third of the book is devoted to linear algebra having comprehensive chapters on vector spaces, linear equations, determinants, quadratic forms and normal forms for matrices.

...

Classic algebra is not the easiest book on undergraduate algebra, but it is a book that those interested in mathematics professionally, as well as the good student, will find valuable and reliable. Furthermore, as P M. Cohn is such a respected writer in the field, it makes sense to have reproduced the previous work in a more accessible single volume form.

**23. Basic algebra: groups, rings and fields (2002), by Paul M Cohn.**

**23.1. From the Publisher.**

Basic Algebra is the first volume of a new and revised edition of P.M. Cohn's classic three-volume text Algebra which is widely regarded as one of the most outstanding introductory algebra textbooks. For this edition, the text has been reworked and updated into two self-contained, companion volumes, covering advanced topics in algebra for second- and third-year undergraduate and postgraduate research students.

In this first volume, the author covers the important results of algebra; the companion volume, Further Algebra and Applications, brings more advanced topics and focuses on the applications. Readers should have some knowledge of linear algebra and have met groups and fields before, although all the essential facts and definitions are recalled.

The coverage is comprehensive and includes topics such as:

- Groups

- lattices and categories

- rings, modules and algebras

- fields

The author gives a clear account, supported by worked examples, with full proofs. There are numerous exercises with occasional hints, and some historical remarks.

**23.2. From the Preface.**

Much of the second and third year undergraduate course in mathematics (as well as some graduate work) was covered by [P M Cohn,

*Algebra*. Vol. 2, Second edition, 1989;

*Algebra*. Vol. 3, Second edition, 1991], now out of print. So I was very pleased when Springer-Verlag offered to bring out a new version of these volumes. The present book is based on both these volumes, complemented by the definitions and basic facts on groups and rings. Thus the volume is addressed to students who have some knowledge of linear algebra and who have met groups and fields, though all the essential facts are recalled here. My overall aim has been to present as many of the important results in algebra as would conveniently fit into one volume. It is my hope to collect the remaining parts of Volumes 2 and 3 into a second book, more oriented towards applications.

Apart from chapters on groups (Chapter 2), rings and modules (Chapters 4, 5 and 6) and fields (Chapters 7 and 11), a number of concepts are treated that are less central but nevertheless have many uses. Chapter 1, on set theory, deals with countable and well-ordered sets, as well as Zorn's lemma and a brief section on graphs. Chapter 3 introduces lattices and categories, both concepts that form an important part of the language of modern algebra. The general theory of quadratic forms has many links with ordered fields, which are developed in Chapter 8. Chapters 9 and 10 are devoted to valuation theory and commutative rings, a subject that has gained in importance through its use in algebraic geometry.

On a first encounter some readers may find the style of this book somewhat concise, but they should bear in mind that mathematical texts are best read with paper and pencil, to work out the full consequences of what is being said and to check examples. The matter has been well put by Einstein, who said: "Everything should be explained as far as possible but no further."

**23.3. Review by: Larry C Grove.**

*SIAM Review*

**45**(3) (2003), 607-608.

This book is the first of two volumes of a rather comprehensive account of contemporary abstract algebra. The main focus is on groups, rings, and fields, but it also includes fairly extensive introductions to lattice theory, category theory, bilinear forms and classical groups, and multilinear algebra. The author has packed a remarkable amount of material into 450 pages of text. There are frequent references to the projected second volume, to be titled

*Further Algebra and Applications*, referred to in the text as FA. ... It is refreshing to note that the author does not shy away from appealing to results from analysis or general topology when they can provide simpler proofs of "purely" algebraic results. An example is the appeal to the Gelfand-Mazur theorem in the proof of Ostrowski's second theorem on absolute values. The text appears to be remarkably clear of misprints or mistakes. I noticed only a few, all of them easily corrected. ... The title of FA refers to applications, but it is not made clear what sort of applications will appear. A wide variety is, of course, possible. All in all this appears to be a carefully thought-out and well-written text, a worthy and serious competitor to the small number of other texts that aim to be as comprehensive.

**23.4. Review by: Gerry Leversha.**

*The Mathematical Gazette*

**89**(514) (2005), 153-154.

It is worthwhile putting this book in a historical context. Professor Cohn originally published his 'Algebra' in two volumes, which appeared in 1974 and 1977; the

*Gazette*review hailed it as likely to become 'a standard reference work' if 'sometimes very condensed.' A second edition of the first volume appeared in 1982; the second volume was then split into two for a new edition in 1989 and 1990, the idea being that volume 2 would be aimed at third year undergraduate students and volume 3 at postgraduates and beyond. The reviewer found these books 'fun to read ... (reflecting) the author's enthusiasm for non-commutative rings and fields'. Then, in 2000, there appeared 'Classic Algebra', a single-volume abridged version of the whole project, covering most of the algebra taught at undergraduate level and some topics from graduate courses.

The latest edition is, according to the jacket, the result of a radical reworking and updating of the original text into two self-contained companion volumes. This is the first of the two, covering the most important theoretical results; the second, 'Further algebra and applications', was reviewed in July 2004. What is not so clear from the cover is that this revision is based largely on the last two volumes of the original set and students who tackle it are assumed to have a basic knowledge of vector spaces, groups, rings and fields. This is clearly not an introduction to algebra for any but the quite outstanding first year undergraduate. It is self-contained, but you will need to hold on tight ...

...

Cohn's lucid and literate style id well-known from his previous work. This is a coherent overview of group, ring and field theory which combines brevity with elegance and authority. It is a pity that there are no solutions, not even hints, to the numerous exercises; readers for whom this is a serious flaw should investigate 'Classic algebra', which is complete in this respect. This is unlikely, however, to make any difference to serious algebraists and departmental libraries for whom this book will be an automatic purchase.

**24. Further algebra and applications (2003), by Paul M Cohn.**

**24.1. From the Publisher.**

Further Algebra and Applications is the second volume of a new and revised edition of P M Cohn's classic three-volume text "Algebra" which is widely regarded as one of the most outstanding introductory algebra textbooks. For this edition, the text has been reworked and updated into two self-contained, companion volumes, covering advanced topics in algebra for second- and third-year undergraduate and postgraduate research students.

The first volume, "Basic Algebra", covers the important results of algebra; this companion volume focuses on the applications and covers the more advanced parts of topics such as:

- groups and algebras

- homological algebra

- universal algebra

- general ring theory

- representations of finite groups

- coding theory

- languages and automata

The author gives a clear account, supported by worked examples, with full proofs. There are numerous exercises with occasional hints, and some historical remarks.

**24.2. From the Preface.**

This volume follows on the subject matter treated in Basic Algebra and together with that volume represents the contents of volumes 2 and 3 of my book on algebra, now out of print; the topics have been rearranged a little, with most of the applications in the present volume, while the basic theories (groups, rings, fields) are pursued further in the earlier book. In any case all parts of volumes 2 and 3 are represented. The whole text has been revised, some exercises have been added and of course errors have been corrected; I am grateful to a number of readers for bringing such errors to my attention.

Chapter 1 presents the basic notions of universal algebra: the isomorphism theorems, free algebras and varieties, with the natural numbers, viewed as algebra with a unary operator as an application, as well as the ultraproduct theorem and the diamond lemma. The introduction to homological algebra in Chapter 2 goes as far as derived functors and global dimension, with the case of polynomial rings and free algebras as an application. Chapter 3, on group theory, discusses some items of general interest and importance (group extensions, Hall subgroups, transfer), but also topics which find an echo elsewhere in the book, such as free groups and linear groups. Chapter 4, on algebras, deals with the Krull-Schmidt theorem, projective covers, Morita equivalence and related matters, but stops short of the representation theory of algebras, which would have required more space than was available. This is followed by an account of central simple algebras (Chapter 5), introducing the Brauer group and crossed products. The representation theory of finite groups in Chapter 6 presents the standard facts on representations and characters and illustrates this work by the symmetric group. The next two chapters return to rings; Chapter 7 presents topics on Noetherian rings such as Goldie's theory, as well as polynomial identities and central polynomials, while Chapter 8 deals with the general density theorem, the various radicals and non-unital algebras. Chapter 9, on skew fields, gives a simplified treatment of the Dieudonné determinant and establishes the existence of 'free fields'. Its proof is based on the specialisation lemma, which is of independent interest.

The final two chapters are applications of a different kind. Chapter 10 is an introduction to block codes, in particular linear codes and cyclic codes, as well as some other kinds. Chapter 11 deals with algebraic language theory and the related topics of variable-length codes, automata and power series rings. In both chapters it is only possible to take the first steps in the subject, but we go far enough to show how techniques from coding theory are used in the study of free algebras.

The text assumes an acquaintance with much of Basic Algebra, to which reference is made in the form 'BA' followed by the section number. Definitions and key properties are usually recalled in some detail, but not necessarily on their first occurrence; the reader can easily trace explanations through the index. As before, there are occasional historical references and numerous exercises, often with hints, though no solutions.

**24.3. Review by: Editors.**

*Mathematical reviews*MR1953987

**(2003k:00001)**.

The author's classic text Algebra first appeared in two volumes [

*Algebra*, Vol. 1, 1974;

*Algebra*. Vol. 2, 1977]. For the second edition [

*Algebra*. Vol. 1, Second edition, 1982;

*Algebra*. Vol. 2, Second edition, 1989;

*Algebra*. Vol. 3, Second edition, 1991], the original Volume 2 was split into Volumes 2 and 3. The three-volume edition has now been reworked and updated into two companion volumes of which the book under review is the second; the first volume has appeared [

*Basic algebra*, 2003].

**24.4. Review by: Robert Curtis.**

*The Mathematical Gazette*

**88**(512) (2004), 381.

Together with the subject matter dealt with in Basic algebra, this volume represents the contents of the second and third volumes of the author's book on algebra, which is now out of print. The original intention for that three volume opus was that the first volume would cover the standard material delivered in the first two years of an undergraduate course in mathematics; the second volume would cover various topics often presented in the final year (it being acknowledged that there is generally more variation between courses at this level), whilst the third volume would be pitched at a level appropriate for advanced undergraduates and postgraduates.

Although the material covered by these two new volumes is unaltered, the author has chosen to make significant changes to the order in which it is presented. Thus all the fundamental concepts and basic theories are found in Basic algebra, whilst Further algebra and applications, as its title suggests, devotes itself to applications of these basic ideas.

In the preface to the first edition the author states:

My aim throughout has been to present the basic techniques of the subject in their simplest form, but to develop them far enough to obtain significant results.This lofty ambition has been achieved to a remarkable degree. The current text is far from being a compendium of important results, which some books covering such a vast area are forced into becoming. On the contrary, the reader is treated to the spectacle of a very fine mathematician developing many diverse branches of algebra in a direct and illuminating manner.

This volume, which is now clearly pitched at a postgraduate or advanced undergraduate level, begins with chapters on universal algebra and on homological algebra. There follow sections devoted to rings, skew fields, algebras and groups, including the representation theory of finite groups. Each of these sections deals with quite sophisticated topics; thus, for instance, the group theory chapter covers extensions of groups, Hall subgroups, the transfer and free groups-topics which you might expect to find in a third year group theory course at university. Sections on the symplectic and orthogonal groups, which were elsewhere in Algebra, have been moved into this chapter. The chapter on representations develops the subject as far as induced representations, and then proves the theorems of Burnside and Frobenius.

A general chapter on algebras, which includes the Krull-Schmidt theorem and Morita equivalence, is followed by another on central simple algebra. Noetherian rings and rings without finiteness assumptions each have chapters devoted to them. ...

In all, this volume contains a gold-mine of algebraic nuggets, each developed in a direct and economical manner highlighting the most important results. It is an invaluable source and is to be strongly recommended.

**25. Free ideal rings and localization in general rings (2006), by Paul M Cohen.**

**25.1. From the Publisher.**

Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalisation, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note.

Last Updated March 2021