1.1. From the Publisher.

The central idea consists of representing rings as rings of endomorphisms of an additive group, which can be achieved by means of the regular representation. This book describes the theory of rings in which both maximal and minimal conditions hold for ideals.

1.2. From the Preface.

See THIS LINK.

1.3. Review by Claude Chevalley.
Mathematical Reviews MR0008601 (5,31f).

The book is mainly concerned with the theory of rings in which both maximal and minimal conditions hold for ideals (except in the last chapter, where rings of the type of a maximal order in an algebra are considered). The contents therefore cover less ground than the title would seem to indicate, since the theory of commutative rings with maximal condition only is entirely omitted. The central idea consists in representing rings as rings of endomorphisms of an additive group, which can be achieved by means of the regular representation.

Chapters I and II establish the fundamental facts about groups with operators and vector spaces (over division rings). Chapter III, which is concerned with the theory of principal ideal rings, is more or less a digression from the general theme of the book. The existence and invariance of elementary divisors for a matrix with coefficients in a principal ideal ring are proved. The whole theory is applied to the study of a semi-linear transformation, including the proof of a generalised "Hauptgeschlechtsatz". Chapter IV contains the structure theory of rings of endomorphisms and its applications to the structure theory of abstract rings. The classical Wedderburn-Artin structure theorems are proved for rings satisfying the descending chain condition; the proof follows the pattern of the improvement made by R Brauer on the Hopkins method. Applications are made to the theory of projective representations of a finite group and of crossed products algebras defined by means of a factor set. Chapter IV ends with the "Galois theory" of a division ring.

The special properties of those rings which are algebras over a field are studied in chapter V. There is not much to add to the results of chapter IV as far as structure theory is concerned, but the notion of direct product of algebras brings with itself a new series of notions and results: Brauer group, representations of an algebra in a division algebra, crossed products. The chapter ends with a discussion of the minimal polynomial of an element in an algebra and the theorem of Wedderburn on the existence of a semi-simple subalgebra which gives a complete set of representatives for the residue classes modulo the radical. Chapter V treats of the ideal theory in a maximal order of a ring satisfying both chain conditions: factorisation theorem for two sided ideals, Brandt groupoid and factorisation of one sided normal ideals.

On the whole, the book is an up to date and very clear exposition of a large portion of modern algebra. The general idea of using representation theory (in a generalised sense) in order to obtain the theorems of structure is followed in a very consequent way throughout the book. The book is quite accessible to a beginner, and provides at each step interesting applications of the general theory. The only criticism of the reviewer would be directed at the omission of the theory of symmetric and related algebras.

1.4. Review by: C C MacDuffee.
Science, New Series 99 (2566) (1944), 182-183.

This is the second book in a new series of expository books entitled "Mathematical Surveys" which is edited and published by the American Mathematical Society. The books of this series are expected to be authoritative and comprehensive within the field covered up to the time of publication. They will be of incalculable value to research mathematicians, who until the war were largely indebted to foreign publishers for such treatises. The present book by Jacobson is a worthy member of this series. It is not, however, recommended to the beginner.

The ring is the present evolutionary form to which linear algebras and hypercomplex systems are ancestral and of which they are special instances. The modern structure theory of linear algebras dates from the publication in 1907 of Wedderburn's thesis, and the structure of rings dates from Artin's paper of 1927. The representation theory of rings and their ideal theory is due to Emmy Noether and many other workers.

The author divides his subject into three parts: structure theory, representation theory and arithmetic ideal theory. In Chapter 1 he lays the foundations of the theory of endomorphisms of a group and throughout the book makes extensive use of the theory of rings of endomorphisms. By using the regular representations, the theory of abstract rings is obtained as a special case of the more concrete theory

of endomorphisms. Moreover, the theory of modules, and hence representation theory, may be regarded as the study of a set of rings of endomorphisms all of which are homomorphic images of a fixed ring.

Chapter 2 deals with vector spaces and Chapter 3 with the arithmetic of non-commutative principal ideal domains. Chapter 4 is devoted to the development of these theories and to some applications to the problem of the representation of groups by projective transformations and to the Galois theory of division rings. The first part of Chapter 5 treats the theory of simple algebras over a general field; the second part is concerned with the theory of the characteristic and minimum polynomials of an algebra and the trace criterion for separability of an algebra.

The book is practically self-contained and embraces in its 150 pages a large amount of factual material. Such conciseness is obtained at the expense of elegance of typography, for many equations which would have looked better in displayed form have been run into the text. But this is a minor criticism of a book which is well planned and executed in a masterly manner.

1.5. Review by: Reinhold Baer.
Bull. Amer. Math. Soc. 52 (3) (1946), 220-222.

In recent years the theory of rings has been one of the centres of most vigorous mathematical life. Although originally an outgrowth of the theory of algebras, it has made itself completely independent of its origin. This was necessitated by two considerations. Firstly, it appeared that the special hypotheses inherent to the theory of algebras were not needed for the greater part of the theory of rings. The latter theory, thus stripped of unnecessary encumbrances, became more general and at the same time simpler, clearer and more elegant. Secondly, there are some important applications of the theory of rings that do not fit into the framework of the theory of algebras, such as the applications to the rings of endomorphisms of abelian operator groups.

In the book under consideration a comprehensive account is given of the theory of rings. In view of the fact that mathematicians from all over the world have contributed toward the growth of the theory and that their results are scattered over all the international mathematical periodicals, the collection of the material is by itself no mean task (Jacobson's bibliography covers eight pages of fine print). But it is even more important and difficult to obtain a unified treatment of such a host of different methods and points of view. In this Jacob-son has succeeded admirably by means of the methodological principle which he uses. The knowledge obtained in the abstract theory of rings is first applied to the study of abelian operator groups over rings (Emmy Noether's representation moduli). The structural theory of abelian operator groups is next used for an investigation of their rings of endomorphisms. The cycle is closed by means of the following theorem which permits the application of the theory of endomorphism rings to the abstract theory of rings: If $R$ is an abstract ring with an identity element, we denote by $a_r$ (by $a_l )$ for $a$ in $R$ the linear transformation which maps the element $x$ in $R$ onto $xa$ (onto $ax$); if we consider the additive group $R_{+}$ of $R$ as an operator group with respect to the $an$, the ring of the $a$, is just the full endomorphism ring and it is essentially the same as the original abstract ring $R$.
...
This very excellent treatment of the theory of rings is more than a compendium. For not only does it offer a résumé of the contents of this theory, but it gives at the same time a very instructive introduction into the working methods used here which will be helpful to others than the algebraists. Since the book is, apart from a very few exceptions, quite self-contained, not much previous knowledge is needed, although it seems desirable to be familiar with the ways of thinking practised in abstract algebra. Thus the book should not only be indispensable to every worker in the theory of rings, but may also be used in connection with an introductory course in abstract algebra.

2.1. From the Preface.

The present volume is the first of three that will be published under the general title Lectures in Abstract Algebra. These volumes are based on lectures which the author has given during the past ten years at the University of North Carolina, at The Johns Hopkins University, and at Yale University. The general plan of the work is as follows: The present first volume gives an introduction to abstract algebra and gives an account of most of the important algebraic concepts. In a treatment of this type it is impossible to give a comprehensive account of the topics which are introduced. Nevertheless we have tried to go beyond the foundations and elementary properties of the algebraic systems. This has necessitated a certain amount of selection and omission. We feel that even at the present stage a deeper understanding of a few topics is to be preferred to a superficial understanding of many.

The second and third volumes of this work will be more specialised in nature and will attempt to give comprehensive accounts of the topics which they treat.

2.2. Review by: Walter Ledermann.
The Mathematical Gazette 36 (318) (1952), 289-290.

In the present volume the basic concepts of algebra are introduced and the general theories of groups, rings, fields and lattices are developed systematically from the foundations. No previous knowledge of the subject is assumed, but at one place the reader is expected to be familiar with the elementary properties of determinants of any order. Although the author does not intend to treat exhaustively any of the topics selected, he carries the investigation beyond the most elementary level because he holds that even at the present stage a "deeper understanding of a few topics is to be preferred to a superficial under-standing of many."

The following brief summary of the contents will give an idea of the ground covered. The Introduction deals with the fundamental notions of sets and mappings. In Chapter I semigroups and groups are defined and the theory is developed as far as the fundamental theorem on homomorphisms. The next three chapters deal with general ring and field theory, including the construction of the field of fractions of a commutative integral domain and a discussion of simple transcendental or algebraic extensions of a field. This is followed by a detailed account of factorisation in a commutative integral domain. The properties of a domain being Gaussian (existence of unique factorisation) or being Euclidean (existence of a Euclidean algorithm relative to a valuation function) or being a principal ideal domain are analysed and their mutual relationship is examined. Chapter V contains the most important results about groups with operators, in particular the homomorphism theorems and the Jordan-Hölder and the Krull-Schmidt theorems. The next chapter begins with an account of modules, a discussion of the chain conditions and a proof of the Hilbert basis theorem. This is followed by an exposition of Noetherian rings and of integral dependence. The final Chapter VII is devoted to lattice theory, with emphasis on applications to group and ring theory.

It is evident that this book stresses the abstract approach to algebra, although here and there some concrete material is included such as symmetric polynomials, quaternions and integers in quadratic fields. The author himself admits that the beginner may find the account "at times uncomfortably abstract" and he urges him to study the supplementary exercises and examples in order to consolidate his newly acquired knowledge. But the exposition is so lucid that even such a beginner, if he heeds the author's advice, should not find it too hard to become familiar with the abstract concepts introduced. He certainly cannot fail to appreciate the elegance and beauty of algebraic structures and admire the simple yet powerful methods which algebraists have devised during the last three decades. With consummate expository skill Prof Jacobson has provided a text-book of abstract algebra suitable for comparatively inexperienced students, who have not previously been acquainted with those concrete facts of classical algebra from which the modern abstract theories have sprung. It is not a bold prediction to affirm that these Lectures will exert a powerful influence on the teaching of algebra to Honours students at our universities. Being thoroughly modern in outlook this work is an excellent preparation for the study of contemporary research papers. The layout is very pleasing and the printing accurate. After reading this book the first thought that will come to the mind of most people, as it did to the reviewer, is the hope that the appearance of the remaining two volumes will not be long delayed.

2.3. Review by: D Rees.
Mathematical Reviews MR0041102 (12,794c).

The author, in his preface, indicates that this book is intended as the introductory volume of a general treatise on abstract algebra, later volumes dealing with vector spaces and with the theory of fields, including valuation theory and Galois theory, being projected. To paraphrase the preface again, the present volume is intended to introduce the reader to the main branches of modern algebra, the treatment of each branch being carried sufficiently far to give the reader an understanding of the ideas underlying each branch. In the opinion of the reviewer, the author has been brilliantly successful in his aim, and the completed work should prove a worthy successor to such earlier works as van der Waerden's "Moderne Algebra", to which, allowing for the passage of time since its publication, the present work is similar in scope. An idea of the ground covered by the present volume is conveyed by the headings of the seven chapters. These, following an introduction to the notations and concepts of set theory and the theory of relations used in abstract algebra, are as follows:
(1) Semi-groups and groups,
(2) Rings, integral domains, and fields,
(3) Extensions of rings and fields,
(4) Elementary factorisation theory,
(5) Groups with operators,
(6) Modules and ideals,
(7) Lattices.

The first three chapters require little comment. The first chapter deals with elementary group theory up to and including the concept of homomorphism, and presents no unusual features. The second chapter fulfils a similar function for the concepts enumerated in its heading, closing with an account of rings of endomorphisms. The third chapter opens with an account of the extension of a ring in a ring with an identity, and of an integral domain to its quotient field, continues with an account of polynomial rings and the elementary theory of field extensions, both transcendental and algebraic, and concludes with an account of polynomials in several variables, including a treatment of symmetric functions, and rings of functions.

The next chapter is concerned with the ideas clustering around the unique factorisation theorem. A departure from customary practise is that the theory is developed for semigroups rather than integral domains, this emphasising the multiplicative nature of the theory. A semigroup is termed Gaussian if the unique factorisation theorem holds in it, and the early part of the chapter is concerned with obtaining conditions, in a form convenient for applications, that a semigroup should be Gaussian. The second half of the chapter is concerned with classes of integral domains which are Gaussian (more accurately, whose multiplicative semigroups of nonzero elements are Gaussian), and it is shown that all principal ideal domains and also all integral domains with a Euclidean algorithm are Gaussian. The chapter concludes with a proof that the property of being Gaussian is preserved under transcendental extension.

The above four chapters comprise what may be considered as the elementary part of the book, the remaining three chapters being on a more advanced level. The first of these, dealing with groups with operators, is essentially a continuation of chapter I, the concepts of that chapter being transferred to groups with operators, and then extended. The theory of homomorphisms is carried on up to the Jordan-Hölder theorem in the form due to Schreier and Zassenhaus. The remainder of the chapter is concerned with the concept of direct product, and contains a detailed account of the ideas underlying the theorem of Krull and Schmidt, which is also proved. The chapter closes with a short account of infinite direct products.

The next chapter opens with an introduction to the theory of modules (i.e., Abelian groups with a ring acting as operator domain), then deals with the chain conditions, as applied to modules and ideals, and this leads naturally to the Hilbert basis theorem for ideals in a ring of polynomials whose coefficients lie in a ring whose left ideals satisfy the ascending chain condition. The second part of the chapter then deals with the ideal theory of Noetherian rings, with proofs of both the uniqueness theorem for the representation of an ideal as the intersection of primary ideals, and of the uniqueness of isolated components. The last section of the chapter is devoted to the study of integral dependence of elements of a ring relative to a subring, a notion generalising the concept of algebraic integer.

The final chapter is a short introduction to the theory of lattices. The main types of lattice are dealt with, including modular lattices, complemented lattices, and Boolean algebras. Certain theorems proved earlier in the book, now appear in their lattice-theoretic form, among them, the Jordan-Hölder theorem. The chapter concludes with an account of the relation between Boolean algebras and a certain type of ring.

2.4. Review by: Anon.
The Military Engineer 45 (305) (1953), 248.

This book is the first of a three-volume series to be published under this general title. It covers "Basic Concepts." Topics covered in this volume of "Basic Concepts," include: semi-groups and groups, rings, integral domains, fields, extension of rings and fields, elementary factorisation theory, groups with operators, modules, ideals, and lattices.

2.5. Review by: Morgan Ward.
Mathematics Magazine 26 (1) (1952), 39-40.

This book, the first of three projected volumes on the subject by Jacobson, is a very good general introduction to modern algebra. After the necessary (mildly Bourbakian) preliminaries, there follow four chapters on semi-groups and groups; rings, integral domains and fields; extensions of rings and fields; elementary factorisation theory. Then follows an interesting chapter on groups with operators, containing among other things Shreier's refinement theorem, the Krull-Schmidt decomposition theorem, chain conditions, direct and sub-direct products. The sixth chapter on modules and ideals gives the Hilbert basis theorem and the Noether decomposition theorems for ideals of a commutative ring with ascending chain condition.

A reader familiar with the classical treatment of modern algebra by van der Waerden, will see that Jacobson has compressed nearly all the group, ring and ideal theory in van der Waerden's two volumes into less than two hundred pages. In addition, many more recent results are included, and the treatment is throughout easy and lucid. The important subject of linear algebra is omitted; it will be treated separately in a second volume.

The book closes with a chapter on lattice theory but treats fully only those parts, directly applicable to group and ring theory in line with Orë's development of the subject.

There are well-selected groups of exercises, remarkably few misprints and no errors that the reviewer could detect.

2.6. Review by: W H Mills.
Bull. Amer. Math. Soc. 58 (1952), 579-580.

This is the first volume of a projected three volume work designed to give a general treatment of abstract algebra. This volume gives a comprehensive introduction to abstract algebra and its basic concepts. The next two volumes will be more specialised in nature. The second one will deal with the theory of vector spaces and the final volume with field theory and Galois theory.

The present volume is well organised and excellently written. A considerable number of exercises are given that vary greatly in difficulty.

After a short introduction on set theory and the system of rational integers the author begins with a preliminary treatment of semigroups and groups. Non-associative binary compositions are discussed briefly. This is followed by a fairly orthodox treatment of rings, integral domains, and fields. The third chapter deals with various types of extensions of rings and fields, such as the field of fractions of an integral domain, polynomial rings, and simple field extensions.

The next chapter contains a discussion of factorisation theory for commutative semi-groups and integral domains. It is shown that the unique factorisation theorem holds for principal ideal domains and for Euclidean domains. Unique factorisation is also discussed for polynomial rings over rings which have unique factorisation.

In Chapter V the author discusses groups with operators, generalising many of the earlier results on groups and group homomorphisms. The isomorphism theorems are proved and this leads up to a proof of the Jordan-Hölder theorem. This is followed by a discussion of the concept of direct product and a proof of the Krull-Schmidt theorem. Infinite direct products are discussed briefly.

The first part of the sixth chapter deals with the theory of modules.

Ascending and descending chain conditions are discussed and a proof is given of the Hilbert basis theorem. The second half of this chapter contains a discussion of ideal theory in Noetherian rings. It is shown that every ideal can be represented as the intersection of primary ideals and two uniqueness theorems are proved about this intersection.

In the final chapter an introduction to the theory of lattices is given. Modular lattices, complemented lattices, and Boolean algebras are treated briefly. A number of results proved earlier in the book are discussed here from a lattice-theoretic point of view-the Jordan-Hölder theorem being proved for modular lattices.

With this volume the author has made an excellent beginning. The completed work should be one of the best general treatments of abstract algebra available.

3.1. From the Preface.

The present volume is the second in the author's series of three dealing with abstract algebra. For an understanding of this volume a certain familiarity with the basic concepts treated in Volume I: groups, rings, fields, homomorphisms, is presup­posed. However, we have tried to make this account of linear algebra independent of a detailed knowledge of our first volume. References to specific results are given occasionally but some of the fundamental concepts needed have been treated again. In short, it is hoped that this volume can be read with complete understanding by any student who is mathematically sufficiently mature and who has a familiarity with the standard notions of modern algebra.

Our point of view in the present volume is basically the abstract conceptual one. However, from time to time we have deviated somewhat from this. Occasionally formal calculational methods yield sharper results. Moreover, the results of linear algebra are not an end in themselves but are essential tools for use in other branches of mathematics and its applications. It is therefore useful to have at hand methods which are constructive and which can be applied in numerical problems. These methods sometimes necessitate a somewhat lengthier discussion but we have felt that their presentation is justified on the grounds indicated. A student well versed in abstract algebra will undoubtedly observe short cuts. Some of these have been indicated in footnotes. We have included a large number of exercises in the text.

3.2. Review by: T Nakayama.
Mathematical Reviews MR0053905 (14,837e).

This is the second of the three volumes dealing with the fundamentals of abstract algebra. The following incomplete list indicates the content: (Finite-dimensional) vector spaces over a division ring; Linear transformations, duality between vector spaces and their conjugate spaces; Cyclic modules, minimum and characteristic polynomials of a matrix, finitely generated modules over a principal ideal domain, elementary divisor theory, endomorphism ring, linear transformations commutative with a given one; Jordan-Hölder and Krull-Schmidt theorems (for vector spaces with a set of linear transformations as operators); Bilinear forms, symmetric and Hermitian scalar products (the latter with respect to an involutorial automorphism of the underlying division ring); Witt's theorem (with amplifications by Pall, Kaplansky and Dieudonné; Euclidean and unitary spaces; Products of vector spaces, tensor spaces; Structure of the ring of (all) linear transformations of a vector space; Infinite-dimensional vector spaces, their Noether bases, dimensionality of the conjugate space (treated by means of the lemmas of Erdös-Kaplansky and Mackey), finite topology for linear transformations, two-sided ideals in the ring of linear transformations, dense rings, the Chevalley-Jacobson density theorem (on irreducible rings of linear transformations) (which generalises Burnside's and Wedderburn's theorems), isomorphism theorems (which generalise Artin's theorem), dual spaces and dense rings containing finite rank transformations. The book adopts basically the abstract conceptual viewpoint, but often uses calculational and constructive methods. Expositions are elegant, and are full of new ideas and techniques even when classical results are dealt with. There are many exercises of much interest.

3.3. Review by: Walter Ledermann.
The Mathematical Gazette 39 (327) (1955), 76-77.

Vectors and matrices were originally intended mainly as tools for co-ordinate geometry in higher dimensions. But they soon became the objects of algebraical investigation in their own right, and the great masters of the past generation introduced and developed the manifold operations of matrix algebra with little reference to geometry. More recently this trend has been reversed, and the underlying geometrical ideas, usually in a more abstract form, are given preference over manipulative algebraical processes. The dominant concepts are now vector spaces and their linear transformations, conjugate spaces and the transposition of a linear transformation, and duality with respect to a bilinear form. The matrix of a linear transformation is a transitory object depending on the fortuitous choice of a vector basis and other conventions, whilst the use of determinants is almost entirely avoided. For the sake of greater generality the domain of scalars is often taken to be a division ring rather than a field, which makes it necessary to distinguish between left and right vector spaces. This point of view leads to some rather surprising results for instance, it is found that if complementary bases are used in the underlying space and its conjugate space then the matrix of the linear transformation and that of its transpose are equal. Again, familiar concepts like the Kronecker product, when analysed abstractly, appear to be much more involved than the formal treatment suggests.

The book opens with a chapter on abstract vector spaces of finite dimension. This is followed by a discussion of linear transformations and their matrices, and of conjugate spaces and the transposition of a linear transformation. Next comes a full treatment of elementary divisors and of various canonical forms of a single matrix. Sets of linear transformations give rise to the definitions of reducibility and decomposability. The chapter on bilinear forms includes an account of Witt's generalisation of the signature of a quadratic form. The concept of a product of vector spaces furnishes an introduction to tensor spaces and their symmetry classes. The structure of the ring of all linear transformations of a finite-dimensional vector space is examined, and the isomorphic mappings between such spaces are expressed in terms of semi-linear transformations. The final chapter is devoted to infinite-dimensional vector spaces and, in particular, to the author's work on dense rings of linear transformations, an interesting topic in which algebra, topology and analysis meet.

There are numerous examples and exercises. The latter include substantial discoveries by distinguished mathematicians and in such cases it would be desirable to give the exact reference, and not only the author's name, since even the most gifted student may be pardoned for failing to obtain the answer unaided.

Two very minor criticisms are that the index might be more complete, and that cross references would be helped by indicating the numbers of chapters and sections at the top of each page.

There is little doubt that the present volume, like its predecessor in this series, will become one of the chief text-books for advanced students of algebra. Professor Jacobson has succeeded in writing a modern work which nevertheless acquaints the reader with the indispensable facts of the relevant classical theories.

3.4. Review by: J Dieudonné.
Bull. Amer. Math. Soc. 59 (5) (1953), 480-483.

Linear algebra is now universally recognised as perhaps the most important tool of the modern mathematician; its concepts and methods, moreover, when properly reduced to their essential features, are among the simplest and most straightforward imaginable. Nevertheless, it is still not uncommon to find graduate students who are totally unfamiliar with some of the fundamental notions of linear algebra, such as, for instance, the theory of duality. This may perhaps be attributed to the scarcity of good textbooks on the subject; if so, the present volume will undoubtedly do much to remedy this situation. Although this is the second part of a work which will ultimately be a treatise covering the whole of what may be called "elementary modern algebra," the author has taken great pains to make the book as self-contained and as easy to read as possible. Indeed, pedagogical intentions are apparent throughout: for instance, definitions which are particular cases of others already given in the first volume are often stated again, sometimes twice or more, and with increasing generality. In addition, the text is accompanied by many well chosen examples and exercises, which provide ample opportunities for the student to test his understanding of the theory.

4.1. From the Preface.

See THIS LINK.

4.2. Review by: Melvin Henriksen.
Mathematical Reviews MR0081264 (18,373d).

The subject of this book is the structure of (non-commutative) rings without finiteness assumptions. Among other things, the majority of the results in the author's "Theory of rings" and in Artin, Nesbitt, and Thrall's "Rings with minimum condition" are generalised, in particular those on semi-simple rings. Certain generalisations of older concepts have played a key role in the emancipation of the structure theory of rings from chain conditions. The author's exploitation of Perlis' characterisation of the radical of a finite dimensional algebra, the replacement of simple rings by primitive rings, and the use of modular (= regular) maximal right ideals instead of idempotents and minimal right ideals are among those essential developments in this book. There have been a large number of contributions made to the structure theory of rings without finiteness assumptions, especially in the last fifteen years. The author concerns himself primarily (but by no means exclusively) with those of Amitsur, Azumaya, Baer, Chevalley, Dieudonné, Kaplansky, Kurosch, Levitzki, McCoy, Nakayama, and himself.
...
The book seems to be addressed primarily to experts in the field or to mathematicians who have a substantial knowledge of the theorems and techniques of modern algebra. The author's statement in the preface "The only knowledge assumed is that of the rudiments of ring and module theory such as is found in any of the introductory texts to abstract algebra." is, at a minimum, misleading. Far too many "well-known" theorems are quoted without reference to make it possible for a student to read it without substantial guidance from a more experienced person.
...
The exposition is complete, but terse. Almost always, the author proceeds from the general to the particular. Motivation is usually given, but often is missing. On the positive side, the material of Chapters VIII, IX, and X is especially well-motivated. On the other hand, the reader unfamiliar with Kronecker products has to wait a while before discovering why they are studied. In the same vein, semisimple and primitive rings are introduced on page 4, but simple rings are not introduced until page 39 (where it is shown that in the presence of the minimum condition for right ideals, a ring is primitive if and only if it is simple).
...
The above criticisms notwithstanding, the author has done an impressive job of gathering together the bulk of the important contributions to the structure theory of rings without finiteness assumptions, a service for which all present and future research workers in this area should be thankful. It is replete with interesting unsolved problems, and will undoubtedly become a standard reference for some years to come. It represents a valuable addition to the Colloquium series.

4.3. Review by: B H Neumann.
The Mathematical Gazette 43 (343) (1959), 73.

Thirteen years have elapsed between the publication of Jacobson's American Mathematical Society Survey Theory of Rings and that of his present American Mathematical Society Colloquium Publication Structure of Rings, and the difference between these two books is a measure of the success of the former. The main progress in ring theory that has been made, largely initiated or supported by the author's earlier survey, and that is the subject of this book, is the freeing of the theory from finiteness assumptions. Among the signal achievements of the theory are results on algebras with polynomial identities and on algebraic algebras that go far towards a complete solution of Kurosh's ring-theoretic analogue of the famous Burnside Problem in the theory of groups.

The author himself is among the main contributors to this development; he is, moreover, a skilful and expert expositor at all levels-witness his Lectures on Abstract Algebra. All this contributes to make the present work an outstanding exposition of present-day ring theory.

An admirable feature of the author's mathematical style is that he always tells the reader what he is going to do, and why. The text reads smoothly, and the mathematical symbols are never allowed to run away with the argument. Little previous knowledge of ring and module theory is assumed; but it is not a beginners' book: the subject matter is so far advanced that the reader has to bring some maturity to the study of the book. He is helped by examples appended to some chapters and references to further work appended to all chapters. There is also a useful bibliography and a full subject index. The printing is good, misprints are few, and the price is up-to-date.

5.1. From the Preface.

The present book is based on lectures which the author has given at Yale during the past ten years, especially those given during the academic year 1959-1960. It is primarily a textbook to be studied by students on their own or to be used for a course on Lie algebras. Besides the usual general knowledge of algebraic concepts, a good acquaintance with linear algebra (linear trans-formations, bilinear forms, tensor products) is presupposed. More-over, this is about all the equipment needed for an understanding of the first nine chapters. For the tenth chapter, we require also a knowledge of the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras.

The subject of Lie algebras has much to recommend it as a subject for study immediately following courses on general abstract algebra and linear algebra, both because of the beauty of its results and its structure, and because of its many contacts with other branches of mathematics (group theory, differential geometry, differential equations, topology). In this exposition we have tried to avoid making the treatment too abstract and have consistently followed the point of view of treating the theory as a branch of linear algebra. The general abstract notions occur in two groups: the first, adequate for the structure theory, in Chapter I; and the second, adequate for representation theory, in Chapter V. Chapters I through IV give the structure theory, which culminates in the classification of the so-called "split simple Lie algebras." The basic results on representation theory are given in Chapters VI through VIII. In Chapter IX the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter X to the problem of sorting out the simple Lie algebras over an arbitrary field.

No attempt has been made to indicate the historical development of the subject or to give credit for individual contributions to it. In this respect we have confined ourselves to brief indications here and there of the names of those responsible for the main ideas. It is well to record here the author's own indebted-ness to one of the great creators of the theory, Professor Hermann Weyl, whose lectures at the Institute for Advanced Study in 1933-1934 were truly inspiring and led to the author's research in this field. It should be noted also that in these lectures Professor Weyl, although primarily concerned with the Lie theory of continuous groups, set the subject of Lie algebras on its own independent course by introducing for the first time the term "Lie algebra" as a substitute for "infinitesimal group," which had been used exclusively until then.

A fairly extensive bibliography is included; however, this is by no means complete. The primary aim in compiling the bibliography has been to indicate the avenues for further study of the topics of the book and those which are immediately related to it.

5.2. Review by: M S Huzurbazar.
Current Science 31 (8) (1962), 351.

In this book the author presents a systematic account of the structure theory of Lie Algebras and their representations. It is meant to be a textbook for a course on Lie Algebras and is based on lectures which the author has given at the Yale University during the past ten years. Besides the usual general knowledge of algebraic concepts, a good acquaintance with linear algebra, elements of Galois theory and Wedderburn structure theory of associative algebras is presupposed. An idea of the coverage of the book may be obtained from the following listing of the chapter headings.

I. Basic concepts,
II. Solvable and Nilpotent Lie Algebras,
III. Cartan's criterion and its consequences,
IV. Split Semi-simple Lie Algebras,
V. Universal Enveloping Algebras,
VI. The theorem of Ado-Iwasawa,
VII. Classification of irreducible modules,
VIII. Characters of the irreducible modules,
IX. Automorphisms,
X. Simple Lie Algebras over an arbitrary field.

A welcome feature of the book is the orientation prefixed to each chapter and a good number of exercises listed at the end of each chapter. These exercises (166 in all) are supplemented with occasional hints and references. An extensive bibliography and a general index add to the facilities of reference and further study. Although no attempt has been made to indicate the historical development of the subject, there are brief indications here and there of the names of those responsible for the main ideas. In short, it is a very valuable book on the subject.

5.3. Review by: R J Crittenden.
Pi Mu Epsilon Journal 3 (7) (1962), 349.

This book gives a very concise, closely knit presentation of the theory of Lie algebras. The objective is to provide a thorough background in the most general context in recognition of the growing relevance of the theory in a number of different fields, such as Lie groups, algebraic groups, and free groups. In particular, the restrictions on the base field are minimal and are introduced only when necessary. For example, the classical classification of semi-simple Lie algebras is extended to split semi-simple Lie algebras, namely, those which have a Cartan subalgebra $H$ such that for every $h \in H$, $h$ has its characteristic roots in the base field. This is a fairly minor point, but it illustrates the spirit in which the author approaches the subject.

The prerequisites, as stated by the author, are a thorough foundation in linear algebra for the first nine chapters, plus some knowledge of Galois theory and the structure theory of associative algebras for the last chapter. This already puts the book out of reach of the average first year graduate student, to say nothing of that intangible, mathematical maturity, clearly necessary for negotiating such a compact work.

Briefly, the first four chapters are concerned with the structure of Lie algebras and the classification of the semi-simple ones. Levi's decomposition theorem and an introduction to the cohomology theory of Lie algebras are included, and Dynkin's method is followed in the classification. The next chapters are concerned with representations of Lie algebras. The universal enveloping algebra is introduced and used to reduce the problem to that of representations of associative algebras, the theorem of Ado-Iwasawa is proved, Cartan's classification of irreducible representations of a semi-simple Lie algebra is given, via modules, and Weyl's formula for the character of such a representation is derived. The ninth chapter studies the automorphism group of a Lie algebra, while in the last chapter the classification of simple Lie algebras over arbitrary fields of characteristic O is investigated. At the end of each chapter there is a section of stimulating, non-trivial problems.

Much of this material is obtainable elsewhere, ultimately in the works of E Cartan, more readably in the volumes of Chevalley on Lie Groups, the Paris "Sophus Lie" seminar notes, Bourbaki, and the notes of H Freudenthal on Lie groups. However, the completeness and generality of the present exposition make it a valuable addition to the literature both as a reference and as a text for an advanced graduate course. Its appearance is sure to be welcomed by many experts in related fields.

5.4. Review by: Rimhak Ree.
Mathematical Reviews MR0143793 (26 #1345).

This book contains a very clear presentation of the theory of Lie algebras, from basic concepts to the deepest part of the theory. It is gratifying to see that the theory on the classification of irreducible modules is now available in such a simplified and elementary form. The author's work on the classification of simple Lie algebras over an arbitrary field is also presented in a unified manner. The book consists of 10 chapters. We shall review each chapter separately.

Chapter I: Basic concepts. Chapter II: Solvable and nilpotent Lie algebras. The theorems of Engel and Lie on nilpotent and solvable Lie algebras of linear transformations and the decomposition into weight spaces of a module are given by considering the enveloping associative algebra of the given linear Lie algebra. A characterisation is given of linear Lie algebras with semi-simple enveloping associative algebras. Chapter III: Cartan's criterion and its consequences. Cartan's criterions on solvability and semi-simplicity in terms of the Killing form are given. The complete reducibility of modules over a semi-simple Lie algebra is proved by using a lemma of Whitehead. The "radical splitting" theorem of Levi and its improvement by Malcev and Harish-Chandra are given. This chapter also contains a representation theory of the split 3-dimensional simple Lie algebra, which is basic in the whole theory of Lie algebras, and the Jacobson-Morozov theorem on completely reducible linear Lie algebras.

Chapter IV: Split semi-simple Lie algebras. These algebras are completely classified. Since the algebras are assumed to be split, the algebraic closedness of the ground field is not needed. Chapter V: Universal enveloping algebra. The existence and basic properties of the universal enveloping algebra, including the Poincaré-Birkhoff-Witt theorem, are proved. Also, free Lie algebra, the Campbell-Hausdorff formula, cohomology theory, restricted Lie algebras of characteristic p, and restricted universal enveloping algebras are discussed. Chapter VI: The theorem of Ado and Iwasawa. The chapter contains a simplification of Harish-Chandra's proof of Ado's theorem, using universal enveloping algebras, and the author's simple proof of Iwasawa's theorem, which yields additional information on the complete reducibility of representations. Chapter VII: Classification of irreducible modules. A surprisingly simple proof is given for Cartan's theorem which establishes a one-to-one correspondence between irreducible modules of a semi-simple Lie algebra and dominant integral forms on a Cartan subalgebra, and similarly for the existence of semi-simple Lie algebra. Basic irreducible modules are exhibited for some simple Lie algebras.

Chapter VIII: Characters of irreducible modules. After deriving Freudenthal's recursion formula for the multiplicities, Weyl's character formula is derived purely algebraically. From this, formulas of Kostant and Steinberg are derived quickly. Chapter IX: Automorphisms. Basic facts about the automorphisms of semi-simple Lie algebras are given, and for some simple Lie algebras the groups of automorphisms are explicitly determined. ... Chapter X: Simple Lie algebras over an arbitrary field. After some ground work, the determination of simple Lie algebras of the types A-D (except for $D_{4}$) over a field $K$ is reduced to that of simple associative algebras over $K$. The case where $K$ is the real field is considered in detail. In spite of this, the reviewer feels that the book would be more useful if it contained a general theory of real semi-simple Lie algebras.

At the end of each chapter a dozen or two well-selected exercise problems are given.

5.5. Review by: I N Herstein.
The American Mathematical Monthly 71 (5) (1964), 571-572.

Since Hochschild (Bull. Amer. Math. Soc. 69 (1963), 37-39) has given such a thorough review of the material in Jacobson's book, I shall not discuss its substance at any length, but shall confine myself instead to the role the book should play both in our graduate education and as a source of learning the subject on the part of a professional mathematician.

As for the subject matter, suffice it to say that Jacobson gives a very complete discussion of the structure theory, representation theory and automorphisms of Lie algebras, with the emphasis almost totally on the split case. One finds a wealth of material covering the usual topics and, very often, a wide range of unusual ones.

Let us return to the question of the impact of the book. To my mind the book fulfils its two purposes-education of the beginner and the experienced one-to perfection. A person knowing nothing about Lie algebras when he picks up the book will, by the time he finishes it, have been brought close to the current research areas in the purely algebraic parts of the theory. (Very little is said about Lie groups or the interrelation of Lie algebras with analysis.) The development is thorough, in many places novel, devoid of twists and gimmicks, and should offer very little difficulty to anyone well-acquainted with linear transformations. It is the only source that I know which starts the reader at absolute scratch, takes him in a coherent, motivated way through Lie algebras, and finally drops him off at a point where he can speak with some intelligence and authority on the subject. The book should be ideally suited for a second year graduate course; in fact it was used as a text at the University of Chicago with what seems to have been great success.

If the book has any drawbacks and which book does not? - the most serious, in my opinion (and what to others may be one of its great virtues) is that there is a certain uniformity and sameness in the approach to many of the theorems. Also there probably should be more stress laid on the innumerable high spots enjoyed by the subject.

Professor Jacobson deserves the thanks of all of us for having made the beautiful subject matter of Lie algebras so readily available to a wide public, both to the sophisticated old-timer and to the budding young student.

5.6. Review by: G Hochschild.
Bull. Amer. Math. Soc. 69 (1) (1963), 37-39.

Although Lie algebras are about 80 years old one might say that they are now barely past their adolescence. They appeared originally as "infinitesimal groups," i.e., as objects derived from Lie groups for the purpose of linearising some of the basic group theoretical problems. However, it soon became evident that the resulting linear problems concerning the structure and the representations of Lie algebras were difficult and of a quite novel type, so that the algebraic technique existing at the turn of the century was inadequate to the tasks thus imposed on it.

It was this very difficulty which led to some of the most beautiful and most fruitful developments in linear algebra, the decisive steps being taken (in this order) by W. Killing, E Cartan and H Weyl (1888-1926). Since the methods used involved analysis and topology, it took several decades more right up to the present to put Lie algebra theory on a firm purely algebraic foundation and to extend the results to base fields other than that of the real or complex numbers. It is in this phase of Lie algebra theory that the author of the present book has played an important role. What he presents here is the first essentially complete and self-contained exposition of the main results on the structure and the representations of Lie algebras which, moreover, embodies many original and substantial simplifications of the theory.

There are ten chapters, each of which begins with a brief description of its content and ends with a series of exercises designed so as to either supplement the theory in certain points or to develop some of the requisite algebraic technique. Knowledge of elementary linear algebra is presupposed and is a sufficient prerequisite for the study of all but the last chapter, where Galois theory and some of the Wedderburn structure theory of associative algebras is needed in addition.
...
In essence, this book is a superbly well organised, clear and elegant exposition of Lie algebra theory, shaped by the hands of a master. It remedies what has been an exasperating deficiency in the mathematical literature for many years.

6.1. From the Preface.

The present volume completes the series of texts on algebra which the author began more than ten years ago. The account of field theory and Galois theory which we give here is based on the notions and results of general algebra which appear in our first volume and on the more elementary parts of the second volume, dealing with linear algebra. The level of the present work is roughly the same as that of Volume II.

In preparing this book we have had a number of objectives in mind. First and foremost has been that of presenting the basic field theory which is essential for an understanding of modern algebraic number theory, ring theory, and algebraic geometry. The parts of the book concerned with this aspect of the subject are Chapters I, IV, and V dealing respectively with finite dimensional field extensions and Galois theory, general structure theory of fields, and valuation theory. Also the results of Chapter III on abelian extensions, although of a somewhat specialised nature, are of interest in number theory. A second objective of our ac-count has been to indicate the links between the present theory of fields and the classical problems which led to its development. This purpose has been carried out in Chapter II, which gives Galois' theory of solvability of equations by radicals, and in Chapter VI, which gives Artin's application of the theory of real closed fields to the solution of Hilbert's problem on positive definite rational functions. Finally, we have wanted to present the parts of field theory which are of importance to analysis. Particularly noteworthy here is the Tarski-Seidenberg decision method for polynomial equations and inequalities in real closed fields which we treat in Chapter VI.

As in the case of our other two volumes, the exercises form an important part of the text. Also we are willing to admit that quite a few of these are intentionally quite difficult.

6.2. Review by: Carl Faith.
Mathematical Reviews MR0172871 (30 #3087).

The chapter headings are: Introduction (algebras, tensor product of algebras and vector spaces, etc.); Chapter I, Finite Dimensional Extensions of Fields (Galois theory, normal bases, Galois cohomology and composites of fields); Chapter II, Galois Theory of Equations (Galois' criterion for solvability of equations by radicals); Chapter III, Abelian Extensions (cyclotomic and Kummer fields, Witt vectors); Chapter IV, Structure Theory of Fields (algebraically closed fields, infinite Galois theory, transcendental fields, Luröth's theorem, derivations of fields, Galois theory of purely inseparable fields); Chapter V, Valuation Theory (real valuations, p-adic numbers, Hensel's lemma, non-archimedean valuations, the Hilbert Nullstellensatz, complete fields, ramification index and residue degree); Chapter VI, Artin-Schreier Theory (ordered fields, formally-real fields, real-closed fields, Sturm's theorem, decision method for an algebraic curve (Tarski-Seidenberg), Artin-Schreier characterisation of real-closed fields).

In this book it is assumed that the reader is familiar with the general notions of algebra and the results on fields of Volume I and with the more elementary parts of Volume II, including a knowledge of prime field, and the construction of simple algebraic and transcendental extensions of a field, and elementary factorisation theory, the basic notions of vector space over a field, dimensionality, linear transformation, linear function, composition of linear transformations, and bilinear form. On the other hand, the deeper results on canonical forms of linear transformation and bilinear forms are not needed.
...
Some features of this book which the reviewer finds stimulating are the following: (1) The large number of excellent exercises which have been culled from many sources in and out of the literature. The author states in the Preface: "... we are willing to admit that quite a few of these [exercises] are intentionally quite difficult." (2) The inclusion of many topics never before published in book form, such as the author's theory of purely inseparable extension of exponent 1 (which is preceded by an excellent introduction to the study of derivations of a field), Artin's solution to Hilbert's 17th problem, the Artin-Schreier theory of real-closed fields (specifically, the determination of those fields having finite codimension in algebraically closed fields as the real-closed fields), the decision method (Tarski and Seidenberg) for the solution of $f(x, y) = 0$ over an algebraically closed field, and many others. Combined with the author's novel approach to Galois theory, these topics make the book appealingly fresh.

As the title suggests, this book is not encyclopaedic. At the end of the book the author suggests some papers for further reading, topics which he evidently wistfully omitted, for example, the Galois theory of commutative rings by Chase, Harrison and Rosenberg, the Amitsur cohomology of fields, or the study of extensions of algebraic number fields with prescribed Galois group by Shafarevich.

6.3. Review by: Walter Ledermann.
The Mathematical Gazette 50 (374) (1966), 429-430.

This is the third and final volume of the author's large-scale treatise on abstract algebra. Whilst the first volume deals with the basic algebraical concepts and the second with linear algebra, the main theme of the third volume is the theory of fields in its various aspects.

As one would expect from the general title of these lectures, the approach to the subject is predominantly abstract. General principles, such as the extension of homomorphism and the Jacobson-Bourbaki correspondence, are presented at an early stage and are then powerfully used in the subsequent theory. Nevertheless, an author of Professor Jacobson's deep mathematical insight and experience would rightly insist that mathematical abstractions must prove their worth by the light they shed on the concrete situations from which they originated. Thus the reader is introduced to the classical problems of Galois theory, the solution of a polynomial equation by radicals, Abelian extensions, Artin's solution of Hilbert's 17th Problem (rationally positive polynomials), and many an item that has justly deserved a place in any standard work on algebra since the time of H. Weber. However, the primary object of the present volume is the exposition of those abstract algebraical theories which form the background to contemporary research. Noteworthy features are the substantial chapters on valuation theory and on the real closed fields, which are of interest to analysts, and the discussion of decision methods for an algebraical curve. The following brief summary of the contents will give a clearer indication of the scope of this volume. Introduction: Extension of homomorphisms, algebras, tensor products. Chapter I: Finite-dimensional extension fields, the fundamental theorem of Galois theory, separability, primitive elements, normal bases, finite fields, trace and norm, Galois cohomology, composites of fields. Chapter II: The Galois group of an equation, solubility by radicals, the general equation of the $n$th degree. Chapter III: Abelian extensions; cyclotomic fields, Kummer extensions, Witt vectors, Abelian $p$-extensions. Chapter IV: Structure theory of fields, algebraically closed fields, infinite Galois theory, transcendency basis, derivations, free composites of fields. Chapter V: Valuation theory; Hensel's lemma, ordered groups, Hilbert's Nullstellensatz. Chapter VI: The Artin-Schreier theory; real closed fields, Sturm's theorem, positive definite rational functions, the Tarski-Seidenberg decision method for an algebraic curve.

The book is written in a somewhat terse, yet lucid, style. It bears the hall-mark of the authority and perception of a highly creative mathematician. Arguments and proofs are normally presented in full detail, but at a pace which demands a considerable degree of concentration and maturity on the part of the reader. Some of the exercises are intentionally difficult and are designed to develop the general theory of the text. This is hardly a book for beginners, but the more advanced student of algebra will find in these lectures an invaluable and, indeed, indispensable source of knowledge and inspiration.

6.4. Review by: I N Herstein.
Bull. Amer. Math. Soc. 73 (1) (1967), 44-46.

This book is the third and last of a series of distinguished books by Jacobson which lay out the foundations of abstract algebra. The first two books have already had a wide acceptance and use both as textbooks and source material. It is obvious that this third volume is destined to play the same role. To this reviewer's mind this volume is by far the best and most unusual of the three. In keeping with the style set in all Jacobson's books, be they elementary or very advanced, this book is chock-full of material, often material that is difficult to find elsewhere. As a consequence the book probably will not be easy to read for a casual student, but a serious student who works at the material will get a large pay-off in return for his efforts. There is perhaps a tendency in the book to state results in a very general form which could perplex a person seeing these things for the first time. But these are minor criticisms when confronted with the many, many pluses that the book enjoys.

As its title implies, the book is a study of fields and of Galois theory. The reader expecting the usual Artin treatment of Galois theory has a surprise in store for him he won't find it here. It is refreshing to see Galois theory handled in a different way. The approach is motivated by the work Jacobson has done on noncommutative Galois theory. Not only does it give the classical commutative field results but it also sets up the machinery to handle the non-commutative situation or to pass on to the kind of work on Galois theory being done now by such people as Chase, Harrison and Rosen-berg. It is rooted in the Jacobson-Bourbaki Lemma which describes the commuting ring of a vector space of endomorphisms in the ring of endomorphisms of the additive group of a field. Exploiting this result, the author goes on to develop the Galois theory. In quick order the usual concepts and results come out. Trace and norm make their appearance and there is a short discussion of crossed products and cohomology theory. Having obtained the general results of Galois theory the author sets himself to applying them to the study of the theory of equations. He gets Galois' criterion for solvability by radicals and applies it to obtain Abel's theorem. However, unfortunately no mention is made of the impossibility of trisection or other classical constructibility theorems.
...
This is some book, a real contribution to the mathematics literature and a golden opportunity for an aspiring algebraist eager to learn. The whole mathematical community should be grateful for such a distinguished addition to its teaching and learning arsenal.

7.1. Review by: Carl Faith.
Mathematical Reviews MR0222106 (36 #5158).

The second edition of this book is stated in the author's preface to be identical with the first, excepting minor corrections, an added bibliography, and three appendices.

Appendix A is devoted to the following topics: (1) Radical and primitivity; (2) Division rings; (3) Algebras with polynomial identity and nil algebras. (1) includes the results of Amitsur on the radical of scalar extensions, and the radical of group algebras.
...
Appendix B gives a proof of Goldie's theorem. ... This represents historically the first representation theorems for arbitrary right Noetherian prime, or simple (or semiprime), rings.
...
Appendix C gives a proof (attributed to P J Higgins) of a theorem of Nagata and Higman stating that over a field of prime characteristic $p$, any nil algebra $A$ of bounded index $n < p$ is nilpotent.

8.1. From the Publisher.

The theory of Jordan algebras has played important roles behind the scenes of several areas of mathematics. Jacobson's book has long been the definitive treatment of the subject. It covers foundational material, structure theory, and representation theory for Jordan algebras. Of course, there are immediate connections with Lie algebras, which Jacobson details in Chapter 8. Of particular continuing interest is the discussion of exceptional Jordan algebras, which serve to explain the exceptional Lie algebras and Lie groups.

Jordan algebras originally arose in the attempts by Jordan, von Neumann, and Wigner to formulate the foundations of quantum mechanics. They are still useful and important in modern mathematical physics, as well as in Lie theory, geometry, and certain areas of analysis.

8.2. Review by: E J Taft.
Mathematical Reviews MR0251099 (40 #4330).

The study of Jordan algebras was begun by Jordan, von Neumann and Wigner in an attempt to formulate the foundations of quantum mechanics in terms of the product $A . B = \large\frac{1}{2}\normalsize (AB + BA)$, instead of the associative product $AB$. Its subsequent algebraic development has been largely due to Albert, Jacobson and McCrimmon. The subject has turned out to have important connections to Lie groups and Lie algebras, geometry, and real and complex analysis. The only previous book on the subject is that of H Braun and M Koecher, which while essentially algebraic in nature, was motivated by questions in analysis. The book being reviewed gives a firm algebraic formulation of the foundation and development of the subject, including the relationship to Lie algebras and geometry, but not to the questions in analysis involving homogeneous cones and Siegel half-spaces. Whereas some of the theory of algebras is developed in the book, the reader is referred to certain results for associative and Lie algebras in earlier books of the author, and there is occasional use of notions from algebraic geometry and algebraic groups. The differential calculus of rational functions and mappings is developed in the text.
...
In conclusion, we have been provided with a superb, extremely complete account of the foundations, techniques and results in a subject whose importance is perhaps only recently starting to be appreciated, written by one of the main developers of the theory. The book is wonderfully organised, from both the mathematical and writing points of view. It is certainly a "must" for anyone who will have any contact with Jordan algebras - either as such, or through geometry or analysis. The author is to be congratulated on his achievement, and thanked for making this comprehensive account of the subject available to the mathematical community.

8.3. Review by: R D Schafer.
Bull. Amer. Math. Soc. 79 (1973), 509-514.

In 1948 the author of this book turned his attention to Jordan rings and algebras. Much of the further development of the subject is due to him, to his students, and to others who have been strongly influenced by him. This book not only incorporates a major portion of the author's research over a twenty-year period, but also is dominated by the fundamental contributions he has made. The book is very carefully organised, and a study of it is indispensable for anyone seriously interested in Jordan algebras.
...
What this sketchy listing of some of the contents of this book fails to convey is the superb organisation of the vast amount of material which the author has included. Representation theory is basic to the entire presentation. It is not only that individual topics are interesting in themselves as they are developed, but their impact is felt consistently throughout the remaining pages. This is a book which one can come back to again and again, gaining new insights every time.

9.1. From the Preface.

In these lectures we shall give a detailed and self-contained exposition of McCrimmon's structure theory including his recently developed theory of radicals and absolute zero divisors which constitute an important addition even to the classical linear theory. In our treatment we restrict attention to algebras with unit. This effects a substantial simplification. However, it should be noted that McCrimmon has also given an axiomatisation for quadratic Jordan algebras without unit and has developed the structure theory also for these.

Perhaps the reader should be warned at the outset that he may find two (hopefully no more) parts of the exposition somewhat heavy, namely, the derivation of the long list of identities in §1.3 and the proof of Osborn's theorem on algebras of capacity two. The first of these could have been avoided by proving a general theorem in identities due to Macdonald. However, time did not permit this. The simplification of the proof of Osborn's theorem remains an open problem. We shall see at the end of our exposition that this difficulty evaporates in the important special case of finite dimensional quadratic Jordan algebras over an algebraically closed field.

9.2. Review by: E J Taft.
Mathematical Reviews MR0325715 (48 #4062).

These notes, based on the author's Bombay lectures of 1969, develop a portion of the theory of quadratic Jordan algebras, which represents a generalisation, due to K McCrimmon, of the theory of linear Jordan algebras of characteristic not two. Most of the linear theory was developed by the author, and an extensive account of that theory is given in his book Structure and representations of Jordan algebras. McCrimmon's axioms and ideas leading to the generalisations were announced in Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1072-1079. The proofs of many of the results in that article are presented for the first time in the author's notes.
...
In conclusion, the author has provided a great service by furnishing a detailed and self-contained account of McCrimmon's structure theory. He nicely distinguishes, by discussions and examples, how the characteristic two theory differs from the linear theory. The reviewer is forced to note, however, that his pleasure in reading these notes was marred by the high frequency of misprints and omissions of both words and symbols. (In a copy subsequently received by the reviewer, an errata section was included that corrects some of these.) The prospective reader should not be deterred, however, from learning about quadratic Jordan algebras from these very readable notes.

10.1. From the Preface.

The main purpose of these notes is to provide a set of models for the exceptional Lie algebras over algebraically closed fields of characteristic 0 and over the field R of real numbers. The models we give are based on the algebras of Cayley numbers (octonions) and on exceptional Jordan algebras. These are valid also for arbitrary fields of characteristic ≠ 2, 3. Our second aim is to give an introduction to the problem of forms of exceptional simple Lie algebras, especially, the exceptional $D_{4}'$s, $E_{6}$'s and $E_{7}$'s. These are studied by means of concrete realisations of the automorphism groups. We omit a discussion of the earlier results on $G_{2}$'s and $F_{4}$'s due respectively to the author and to Tomber. Further developments along these lines are due to the author (for $D_{4}$'s), to Allen (for $D_{4}$'s), to Ferrar (for $E_{6}$'s and $E_{7}$'s). The papers containing these results are given in the Bibliography.

The first half of these notes (§1-7) were written in 1958. §8 was written in 1965 and most of this has appeared in Ferrar's paper. The remainder of the notes were written in 1970. Section 11 on the Killing forms is the result of joint work with Ferrar.

10.2. Review by: N R Wallach.
Mathematical Reviews MR0284482 (44 #1707).

In these notes the author gives explicit techniques for the construction of the exceptional simple Lie algebras ($E_{6}, E_{7}, E_{8}, F_{4}$ and $G_{2}$) over a field of characteristic 0 (actually fields of characteristic ≠ 2). Using this construction he also finds the real forms of these algebras. The basic technique is to realise the algebras relative to their actions on exceptional Jordan algebras and/or Cayley algebras (e.g., $G_{2}$ is the derivation algebra of the Cayley numbers, and $F_{4}$ is the derivation algebra of a 27-dimensional Jordan algebra). Interestingly enough, $D_{4}$ is classed as an exceptional Lie algebra. This is reasonable, due to the "mystery" of triality (the 8-dimensional spin representations (2 of them) and the ordinary 8-dimensional representation of $D_{4}$ are infinitesimally a matter of choice; once the choice is made then one corresponds to $SO(8)$ and the other two correspond to $Spin(8)$).

These notes are a reasonable addition to a library that contains the author's book Lie algebras (1962). In fact, reading the notes to understand pages 142-146 of Lie algebras is worth their (high) price.

11.1. From the Preface.

The present book, Basic Algebra I, and the forthcoming Basic Algebra II were originally envisioned as new editions of our Lectures in Abstract Algebra. However, as we began to think about the task at hand, particularly that of taking into account the changed curricula in our undergraduate and graduate schools, we decided to organise the material in a manner quite different from that of our earlier books: a separation into two levels of abstraction, the first - treated in this volume - to encompass those parts of algebra which can be most readily appreciated by the beginning student. Much of the material we present here has a classical flavour. It is hoped that this will foster an appreciation of the great contributions of the past and especially of the mathematics of the nineteenth century. In our treatment we have tried to make use of the most efficient modern tools. This has necessitated the development of a substantial body of foundational material of the sort that has become standard in textbooks on abstract algebra. However, we have tried throughout to bring to the fore well-defined objectives which we believe will prove appealing even to a student with little background in algebra. On the other hand, the topics considered are probed to a depth that often goes considerably beyond what is customary, and this will at times be quite demanding of talent and concentration on the part of the student. In our second volume we plan to follow a more traditional course in presenting material of a more abstract and sophisticated nature. It is hoped that after the study of the first volume a student will have achieved a level of maturity that will enable him to take in stride the level of abstraction of the second volume.

11.2. Review by: James Wiegold
The Mathematical Gazette 60 (411) (1976), 79-80.

I do not doubt that the hope expressed in the last sentence is not a vain one. Genuine beginning students will benefit from this text in that definitions and the most basic results are given in some detail, as well as hosts of interesting and significant examples of the various structures, etc. that arise; it is nevertheless the case that the speed and terseness with which the more advanced topics are covered are pretty unusual for texts of this kind. For the really talented beginner, the book is truly excellent: it is rich in content, novel in approach, and is laced with interconnections and historical detail. A vast provision of exercises for the reader includes some that are reasonably elementary, others that are really quite hard. Many important concepts and results appear as exercises: to name but a few, wreath products of groups and the Sylow subgroups of symmetric groups, the ideal-theoretic form of the Chinese Remainder Theorem, and even a characterisation of the symmetric group of degree five via centralisers of involutions! It is a shame that no indication is given of the difficulty of exercises, since the hardest would defeat any but the most tenacious - a depression-inducing situation if you do not know that questions are supposed to be hard.

Some idea of the scope of the material covered can be obtained from the chapter headings: Monoids and groups; Rings; Modules over a principal ideal domain; Galois theory of equations; Real polynomial equations and inequalities; Metric vector spaces and the classical groups; Algebras over a field; Lattices and Boolean algebras.

A detailed reading of this book and a determined attack on its exercises would repay most mathematicians beyond the level of first-year honours. How excellent it would be if every honours graduate in mathematics were familiar with its methods and material! The appearance of this text is greatly welcome, as will that of the sequel unquestionably be.

11.3. Review by: M F Smiley.
Mathematical Reviews MR0356989 (50 #9457).

This is the first volume of a two-volume text for courses in algebra at the collegiate level. It is based, in part, on the author's books [Lectures in abstract algebra, Vol. I: Basic concepts, 1951; Vol. II: Linear algebra, 1953; Vol. III: Theory of fields and Galois theory, 1964], but it is by no means a mere revision of these books. A very large amount of new material is introduced in order to help the serious elementary student appreciate the foundational abstractions by exploiting their connections with classical mathematical problems.

The guiding principles of the author in preparing this volume are forcefully stated as follows: "Much of the material we present here has a classical flavour. It is hoped that this will foster an appreciation of the great contributions of the past and especially of the mathematics of the nineteenth century. In our treatment we have tried to make use of the most efficient modern tools. This has necessitated the development of a substantial body of foundational material of the sort that has become standard in textbooks on abstract algebra. However, we have tried throughout to bring to the fore well-defined objectives which we believe will prove appealing even to a student with little background in algebra."
...
All of this is given us with an intensification of the author's well-known expository power. This reader was struck especially by the vast number of historical notes that are cleverly interwoven into the text and are of immense value in sustaining interest. The exercise sets are well constructed to provide some that are easy enough for all students, many which point out mild extensions of the basic theorems and quite a number that will challenge even the best students.

The author looks forward to "an upgrading of college mathematics" despite the clear evidence of the recent and serious deterioration of all collegiate work. This volume will be an indispensable tool for those seeking to reverse this trend.

11.4. Review by: I N Herstein.
American Scientist 63 (6) (1975), 718.

All of Jacobson's books, whether advanced or elementary, as is the book under review, have a common characteristic: they are full of meat. Basic Algebra 1 is designed for the student's first introduction, say at the junior-senior level, to abstract algebra. Of all Jacobson's books, the student will find this text the most readable; it develops its subject in a leisurely, thorough manner, and with loving care. The problem sets are first-rate. The student should find the book a delightful experience.

In a short review, it is difficult to get across the flavour of the book, which has a great deal of substance. Let me try to point out a few features that are unique for a book at this level. To begin with, there is a very thorough treatment of quadratic forms, much beyond the usual treatment of Sylvester's law of inertia. Here we find Witt's Cancellation Theorem and a great deal about orthogonal groups

Perhaps the most unusual feature is the brief development of alternative algebras, Lie and Jordan algebras, and, especially, of the relationship between alternative algebras and composition algebras. This last topic, which was originally in a paper of Jacobson's, is a nice application of linear algebra. He pushes it through to characterise composition algebras and from this derives the classical, beautiful theorem of Hurwitz that sums of n squares compose bilinearly if and only if $n$ = 1, 2, 4, and 8.

12.1. From the Foreword.

These are lecture notes for a course on ring theory given by the author at Yale, September - December, 1973. The lectures had two main goals: first, to present an improved version of the theory of algebras with polynomial identity (over a commutative coefficient ring) based on recent results by Formanek and Rowen and second, to present a detailed and complete account of Amitsur's construction of non-crossed product division algebras.

12.2. Review by: E Formanek.
Mathematical Reviews MR0369421 (51 #5654).

These lecture notes are based on a one-quarter course given by the author. As the title indicates, the notes are an introduction and the range of topics is limited. However, the treatment is up-to-date and anyone who has read the first chapter of these notes should be ready for most PI-papers in the literature.

There are two chapters. The first treats the basic structure theory of PI-rings - Kaplansky's theorem, the Amitsur-Levitzki theorem, central polynomials, and Posner's theorem and generalisations. A few basic results on nil ideals are included.

While the first chapter is of a very general nature, the second chapter is devoted to a specific (and very important) application of PI-theory, Amitsur's proof of the existence of finite-dimensional division algebras that are not crossed products. One part of Amitsur's proof is the construction of a certain "generic" division algebra. This is very much dependent on PI-theory. The remainder of the proof is the explicit construction of division algebras with certain properties. Unfortunately (from a didactic viewpoint) this is not particularly related to PI-theory and requires introducing valuative theory and Laurent series fields.

The prerequisites are modest, except for the field theory in the second chapter. The only overlap with C Procesi's book [Rings with polynomial identity, 1973] is in the fundamental structure theorems, and the use of central polynomials makes the presentation of some of these theorems simpler and clearer here. Procesi's book does not treat noncrossed products, while the topics that form the main thrust of Procesi's book are not treated here: representation theory, invariants, finiteness conditions, and Artin's characterisation of Azumaya algebras of constant rank.

13.1. From the Preface.

This volume is a text for a second course in algebra that presupposes an introductory course covering the type of material contained in the Introduction and the first three or four chapters of Basic Algebra I. These chapters dealt with the rudiments of set theory, group theory, rings, modules, especially modules over a principal ideal domain, and Galois theory focused on the classical problems of solvability of equations by radicals and constructions with straight-edge and compass.

Basic Algebra II contains a good deal more material than can be covered in a year's course. Selection of chapters as well as setting limits within chapters will be essential in designing a realistic program for a year. ...
...
Aside from its use as a text for a course, the book is designed for independent reading by students possessing the background indicated. A great deal of material is included. However, we believe that nearly all of this is of interest to mathematicians of diverse orientations and not just to specialists in algebra. We have kept in mind a general audience also in seeking to reduce to a minimum the technical terminology and in avoiding the creation of an overly elaborate machinery before presenting the interesting results. Occasionally we have had to pay a price for this in proofs that may appear a bit heavy to the specialist.

Many exercises have been included in the text. Some of these state interesting additional results, accompanied with sketches of proofs. Relegation of these to the exercises was motivated simply by the desire to reduce the size of the text somewhat. The reader would be well advised to work a substantial number of the exercises.

An extensive bibliography seemed inappropriate in a text of this type. In its place we have listed at the end of each chapter one or two specialised texts in which the reader can find extensive bibliographies on the subject of the chapter. Occasionally, we have included in our short list of references one or two papers of historical importance. None of this has been done in a systematic or comprehensive manner.

13.2. Review by: M F Smiley.
Mathematical Reviews MR0571884 (81g:00001).

This is a text for a second course in algebra based on the author's earlier book [Basic algebra, I, 1974]. A tremendous amount of algebra has been exposed in the years since the author's Lectures in abstract algebra [Vol. I, Basic Concepts, 1951; Vol. II, Linear algebra, 1953; Vol. III, The theory of fields and Galois theory, 1964] appeared. Naturally no text could accommodate more than a small fraction of these discoveries.

After a solid grounding in category theory and universal algebra, the author presents a selection of the more important advances under the following subdivisions: Modules, Basic structure theory of rings, Classical representation theory of groups, Elements of homological algebra with applications, Commutative ideal theory: General theory and Noetherian rings, Field theory, Valuation theory and Dedekind domains. The text closes with a short chapter on formally real fields. As the titles of some of the subdivisions reveal, there is much more information currently available. Some of this is presented, in the usual way, as exercises for the reader.

This book, in its entirety, is not intended as a text even for a second course in algebra covering a full year. Rather the student and his teacher (if any) must choose some of the eight major subdivisions. The author has presented the topics so as to accommodate such a variety of uses. The writing is very clear, unhurried and completely uncompromising as to the reader's full understanding of the desired results. This method precludes shortcuts at the expense of bewilderment for the reader. Although it entails far more writing, it makes the advances of algebra more readily available to the host of mathematical students for whom algebra has become an indispensable tool rather than the centre of interest.

For the purpose of this review, a detailed analysis of the author's treatment of the eight major subdivisions seems inappropriate. The reviewer will try to discuss only the chapter on representation theory of finite groups, a topic which has justified a number of excellent mathematical texts. The representation spaces have finite dimension, mostly over the field of complex numbers. The complete reducibility of any representation (Maschke) is proved first. The author adds A H Clifford's theorem on the restriction of an irreducible representation to a normal subgroup. The analysis of the group algebra is then presented. Next come the Young diagrams which yield the irreducible representations of the symmetric group. Character theory starts with the orthogonality relations and the computation of the character table. The characters of direct products are computed and applied to (finite) abelian groups. The author proves the famous theorem of Burnside: "A group of order $p^{a}q^{b}$ is solvable if $p$ and $q$ are primes." This theorem remains a challenge in that no proof which avoids representation theory and is as simple as Burnside's has been found. Representations of $G$ induced by representations of subgroups $H$ of $G$ are fully treated, leading to the reciprocity theorem of Frobenius and proceeding to Mackey's decomposition theorem and Brauer's theorem.

The discussion is concluded with Brauer's theorem on splitting fields, the Schur index and a fascinating treatment of Frobenius groups which yields, with the aid of character theory, a variant of a classical proof of the quadratic reciprocity theorem. All of the other subdivisions are written with comparable regard for the illuminations which the last decades have provided. The necessary abstractions are not avoided, but the driving motive of the whole volume is to convince the reader that algebra is alive and living in almost every citadel of the world of mathematics. This book will remain a model of the exposition of the beauty and usefulness of algebraic thinking for many years.

14.1. From the Preface.

This monograph is an expanded version of the lectures on the structure theory of Jordan algebras given by the author at Yale in January - April 1981. An outline of the results was presented in five lectures at a conference on Jordan algebras at the University of Arkansas, April 2-4, 1981. The term "Jordan algebra" is used here as an abbreviation of "unital quadratic unital quadratic Jordan algebra". The structure theory of this class of algebras was presented in our monograph, Lectures on Quadratic Jordan Algebras, published by the Tata Institute of Fundamental Research in 1969. Some of the same ground is covered in the present volume and occasionally we have referred for proofs to the earlier monograph and to our still earlier book Structure and Representations of Jordan Algebras (1968).

Since the appearance of the Tata notes there has been considerable progress in the Jordan structure theory and its applications. Particularly noteworthy is the extension of the theory to Jordan triple systems and Jordan pairs due to K Myberg and O Loos. We shall not touch this aspect of the theory in this volume. The framework we have adopted of unital quadratic Jordan algebras is the same as that of the Tata notes.

Within the more restricted framework of the linear theory (or, equivalently, quadratic Jordan algebras over rings containing $\large\frac{1}{2}\normalsize$) some truly remarkable results have been obtained during the past two years by a young Russian mathematician, E I Zelmanov. A substantial part of these have been extended to the general quadratic case by K McCrimmon. The main goal of the Yale and Arkansas lectures was to present the results of Zelmanov and their extension due to McCrimmon. In the first seven chapters of this book we consider the quadratic theory. However, in the last chapter dealing with Jordan division, we have had to revert to the linear theory. It remains an open problem to remove this restriction from the theory. We are greatly indebted to McCrimmon, first, for sending us preprints and for many discussions on the recent developments and second, for carefully reading an earlier version of this manuscript and offering numerous suggestions for improvements. The numerous explicit references to him do not adequately reflect the extent of his contributions to this monograph.

14.2. Review by: Michel L Racine.
Mathematical Reviews MR0634508 (83b:17015).

Jordan algebras were introduced in 1932 by P Jordan. Shortly thereafter, Jordan, von Neumann and Wigner classified the formally real finite-dimensional Jordan algebras over the reals. All but one of the algebras they obtained were easily seen to be imbeddable in appropriately chosen associative algebras (the Jordan product being given by $A . B = \large\frac{1}{2}\normalsize (AB + BA)$. Such algebras are termed special. Albert then showed that this was not the case for the remaining algebra $H(M_{3}(\mathbb{C})$), the symmetric elements of the 3 × 3 matrices with entries in the Cayley numbers with conjugate transpose involution, thereby kindling the interest of algebraists. By the early 50s a structure theory for finite-dimensional (linear) Jordan algebras over a field of characteristic not 2, closely paralleling the associative theory, had been obtained, mostly by Albert and Jacobson.

This set of notes is the author's third book-length work on Jordan algebras. Each one corresponds to a different major improvement in the development of the structure theory of Jordan algebras. In the author's first book on Jordan algebras [Structure and representations of Jordan algebras, 1968], he developed his theory of linear Jordan algebras over a field of characteristic not 2 satisfying DCC on inner ideals, and the classical theory of finite-dimensional algebras was obtained as a special case. McCrimmon then introduced the concept of a quadratic Jordan algebra over an arbitrary (unital commutative associative) ring of scalars and extended the structure theory to this setting. This was the topic of the second book by the author [Lectures on quadratic Jordan algebras, 1969]. In the book under review, the author presents recent results of E I Zel'manov on prime Jordan algebras and on Jordan division algebras, and the extension of the first set of results to the quadratic setting by McCrimmon. These notes are based on a course given at Yale and on a series of lectures delivered at the University of Arkansas.

15.1. From the Preface.

Since the publication of the first edition a number of teachers and students of the text have communicated to the author corrections and suggestions for improvements as well as additional exercises. Many of these have been incorporated in this new edition.

Two important changes occur in the chapter on Galois theory, Chapter 4. The first is a completely rewritten section on finite fields. The new version spells out the principal results in the form of formal statements of theorems. In the first edition these results were buried in the account, which was a tour de force of brevity. In addition, we have incorporated in the text the proof of Gauss's formula for the number N(n, q) of monic irreducible polynomials of degree n in a finite field of q elements. In the first edition this formula appeared in an exercise. The second important change in Chapter 4 is the addition of Section 4.16, "Mod $p$ Reduction", which gives a proof due to John Tate of a theorem of Dedekind on the existence of certain cycles in the Galois permutation group of the roots of an irreducible monic polynomial$f(x)$ with integer coefficients that can be deduced from the factorisation of $f(x)$ modulo a prime $p$. A number of interesting applications of this theorem are given in the exercises at the end of the section.

In Chapter 5 we give a new proof of the basic elimination theorem. The new proof is completely elementary, and is independent of the formal methods developed in Chapter 5 for the proof of Tarski's theorem on elimination of quantifiers for real closed fields. Our purpose in giving the new proof is that Theorem 5.6 serves as the main step in the proof of Hilbert's Nullstellensatz given in the author's Basic algebra, II. The change has been made for the convenience of readers who do not wish to familiarise themselves with the formal methods developed in Chapter 5.

At the end of the book we have added an appendix entitled "Some topics for independent study", which lists 10 such topics. There is a brief description of each, together with some references to the literature.

15.2. Review by: Haya Freedman.
The Mathematical Gazette 70 (452) (1986), 170.

The first edition of this book was published in 1974. It has been well received and extensively reviewed (see for example, the March 1976 Gazette or Mathematical Reviews 50 (1975) #9457). Except for a few changes the contents of this edition is identical to that of the 1974 one, so the reader can consult the earlier reviews. I would endorse James Wiegold's opinion: "For the really talented beginner, the book is truly excellent."

Now to the changes: the second edition has been completely reset; the layout, which was very pleasant in the first edition, has been further improved, and a special symbol has been introduced to indicate the end of each proof. Some misprints have been corrected, and new ones created. The number of the exercises has been increased by the addition of many stimulating problems; some of the existing exercises have been modified or deleted.

The main changes in the text occur in chapters 4 and 5, and the appendix at the end is new. Section 4.13 on finite fields has been revised, and the proof of Gauss' formula for the monic irreducible polynomials with coefficients in a finite field, incorporated in the text. (In the first edition this formula appeared only in an exercise.)

Section 4.16 on "Mod $p$ Reduction" is completely new. It contains Dedekind's theorem on the structure of the Galois group of an irreducible monic polynomial $f(x)$ with integer coefficients, which structure can be deduced from the factorisation of $f(x)$ modulo a prime $p$. The proof given in the text is based on a beautiful theorem due to John Tate. The reach of Dedekind's theorem is illustrated by using it to determine, in a most elegant way, the Galois group of a given integer polynomial of degree six. These additions are very welcome in view of the growing interest in finite fields and irreducible polynomials, brought about by important applications outside algebra, e.g. to coding theory and to cryptography.

The appendix, entitled "Some topics for independent study" lists 10 such topics. There is a very brief description of each, accompanied by references to the literature. Most of the topics cover beautiful mathematics created by great mathematicians. They range from classical problems, like Hilbert's irreducibility theorem or Schur's work on "The Galois groups of some classical polynomials", to Euclidean rings and non-commutative principal ideal domains. I have just two reservations regarding the appendix: one is that most of the references given are to books, whereas students benefit considerably more from studying original papers than from reading books. The other one is that some of the material would not be easily accessible to undergraduates, and should be supplemented by prerequisites. For example, Motzkin's paper "The Euclidean algorithm" makes more sense when read in conjunction with Wilson's paper "A principal ideal ring that is not a Euclidean ring", Mathematics Magazine 46 (1973).

15.3. Review by: Kenneth Bogart.
The American Mathematical Monthly 92 (10) (1985), 743-745.

This review covers both Vol I (BAI) and Vol II (BAII).

In its two volumes, Basic Algebra comprises an entire education, advanced undergraduate and graduate, in algebra, an education deep enough to serve as a springboard to research in most traditional areas of algebra. It is unlikely that many graduate students, even those specialising in algebra, have learned such a breadth of ideas in such depth. To my knowledge, only Bourbaki has attempted as deep a survey of algebra as appears in BAII. (Of course in their encyclopaedic way, the Bourbaki group covers even more.) Though linear and multilinear algebra is woven throughout BAI and BAII, it does not play a prominent role ... With that exception, BAII takes us on a tour of "core" algebra including commutative algebra, the structure theory of rings and algebras, representation theory for groups, and field theory in addition to its category theory, universal algebra and homological algebra. (Homological algebra, which arose from work on chains and cochains in algebraic topology, is based on the study of sequences of modules connected by homorphisms. Homological algebra has provided proofs of results of algebraic interest, some appearing in BAII, as well as results of topological interest.

Of course, it is unfair to expect a textbook to contain a complete compendium of results; however, I admit to disappointment that after developing the machinery needed to prove the unique factorisation theorem for regular local rings, BAII does not include it even as a series of exercises. To a lesser extent, I felt a similar unsatisfied feeling when the results on modular and distributive lattices were not used in the appropriate places in ring theory (for example, a Dedekind domain is a domain whose lattice of ideals is distributive, and the modularity of the lattice of submodules of a module has not been exploited). When defining a coalgebra, Professor Jacobson comments that "It is nice to have a pretty definition, but it is even nicer to have some pretty examples." His examples are undeniably pretty, but in neither one do we learn what makes the coalgebra structure of interest. Frequently, the examples leave something to our imagination in this way. Since there is a trade-off between the amount covered and the details included in a book, we should not regard the rather telegraphic style of examples in BAII as a defect. However, even the experienced mathematician can profit from discussions of BAII with a more knowledgeable colleague.
...
For graduate students, one might use portions of BAI in a one semester course ... which could be profitably followed by a course from BAII. How long would the course last? As long as possible! Realistically, there are at least four semesters worth of work in BAII. This volume is Professor Jacobson's gift to those of us who use algebra; I believe that as many mathematicians as possible should be exposed to as much of it as possible.

15.4. Review by: Andy R Magid.
The American Mathematical Monthly 93 (8) (1986), 665-667.

For traditional Algebra, Jacobson's Basic Algebra I is hard to beat. It's clear and complete, and it studs the path of groups through rings to fields and beyond with all sorts of mathematical jewels, such as the characterisation of symmetric polynomials, or the identification of the composition algebras. Topics covered include Sylow Theorems, free groups, Euclidean Domains, quaternions, finitely generated modules over PID's, ruler and compass constructions, Galois theory, trancendance of $e$ and $\pi$, real fields, classical groups, simplicity of $PSL_{n}$, real division algebras, and the fundamental theorem of projective geometry. There's no doubt that this is basic Algebra, and all mathematicians need to know it. Moreover, they can learn it from Basic Algebra I: there is attention here to exposition and detail that students (and their teachers) will appreciate.

Mathematical developments in Algebra since the first edition of 1974 - such as the classification of the finite simple groups, the freedom of projective modules over polynomial rings, or the finite generation of groups of integral points on Abelian varieties - are not in this text (properly they would be much closer to the topics of the companion volume Basic Algebra II), but some changes have been made, improving the chapters on Galois theory and Elimination theory. The text was completely reset for this edition, with the attendant risks (although I only found one new typo: the inequality sign in the statement of Theorem 3.9 of Chapter 3 has been incorrectly printed as an equality for this edition). The publisher's blurb which accompanied my review copy says that "a solutions manual accompanies the text". I haven't seen it, and in fact my only evidence for its existence is this single inscription, but this seems like such a wonderful idea that the publisher and author and whoever else may be responsible for this innovation deserve high praise. Would that every text at this level would be so accompanied. (Perhaps also we should comment on the level of this text: the author lists only linear algebra as prerequisite; when we use this book as a text at my university, it is for our first year graduate course which has a year of undergraduate algebra as a prerequisite.)

So what is Algebra and how much should one know? Basic Algebra I is an excellent answer.

16.1. From the Preface.

The most extensive changes in this edition occur in the segment of the book devoted to commutative algebra, especially in Chapter 7, Commutative Ideal Theory: General Theory and Noetherian Rings; Chapter 8, Field Theory; and Chapter 9, Valuation Theory. In Chapter 7 we give an improved account of integral dependence, highlighting relations between a ring and its integral extensions ("lying over," "going-up," and "going-down" theorems). Section 7.7, Integrally Closed Domains, is new, as are three sections in Chapter 8: 8.13, Transcendency Bases for Domains; 8.18, Tensor Products of Fields; and 8.19, Free Composites of Fields. The latter two are taken from Volume III of our Lectures in Abstract Algebra (D Van Nostrand 1964; Springer-Verlag, 1980). The most notable addition to Chapter 9 is Krasner's lemma, used to give an improved proof of a classical theorem of Kurschak's lemma (1913). We also give an improved proof of the theorem on extensions of absolute values to a finite dimensional extension of a field (Theorem 9.13) based on the concept of composite of a field considered in the new section 8.18.

In Chapter 4, Basic Structure Theory of Rings, we give improved accounts of the characterisation of finite dimensional splitting fields of central simple algebras and of the fact that the Brauer classes of central simple algebras over a given field constitute a set - a fact which is needed to define the Brauer group Br(F). In the chapter on homological algebra (Chapter 6), we give an improved proof of the existence of a projective resolution of a short exact sequence of modules.

A number of new exercises have been added and some defective ones have been deleted.

16.2. Review by: P M Cohn.
Mathematical Reviews MR1009787 (90m:00007).

The first edition of this work represented a judicious selection from the ever-growing corpus of "basic algebra". In the new edition there are minor additions (some 20 pages, mainly on composites of fields), but many small changes help to clarify the exposition. This volume presents an account of rings, fields, group representations and homological algebra. As in any comprehensive treatment, one finds a reference here to all the important notions in the subject: rings are taken as far as central simple algebras and the Brauer group; Galois theory (already dealt with in Vol. I is looked at from a more advanced point of view, via the Jacobson-Bourbaki correspondence, and there are separate chapters for valuations and ordered fields. Homological algebra goes up to Ext and Tor and proves Hilbert's syzygy theorem; representation theory includes Brauer's characterisation of characters and the Schur index. But beyond the expected topics, there are many lesser known items which add interest and spice. Examples are ultraproducts, Witt vectors, unirational extensions and (parts of) Pfister theory. Exercises and historical notes help to confirm this volume as a standard algebra text.

16.3. Review by: Kenneth Bogart.
The American Mathematical Monthly 92 (10) (1985), 743-745.

This review covers both Vol I (BAI) and Vol II (BAII). See 15.3 above.

16.4. Review by: F Gerrish.
The Mathematical Gazette 74 (470) (1990), 403-404.

Basic algebra II contains material suitable for a second university-level course in algebra, presupposing a background like that in the early part of Volume I (namely, the elementary theory of sets, groups, rings and modules, and perhaps some Galois theory). A one-year course can be based on it in many ways by selection of topics. The book can also be used for independent reading, and includes much of interest not only to specialists in algebra but to all mathematicians.

Building upon a foundation carefully laid in the first volume, the second begins with some further set theory, treats basic category theory and universal algebra (respectively the general study of mathematical objects and their morphisms, and of algebraic systems) - although ideas from these disciplines have already appeared in Volume I, for example the use of universal characterisations from the former, lattice theory from the latter - and continues the study of modules. Much of this could still be advanced undergraduate work, together with which could come Chapter 6 on elementary homological algebra (the general study of sequences of module morphisms), even though preceded by extensive chapters on structure theory of rings and the classical representation theory of finite groups. These two chapters and the remaining material seem more suitable for postgraduate study. The titles of Chapters 7-11 are: Commutative ideal theory, Field theory, Valuation theory, Dedekind domains, Formally real fields. Some of the chapters are virtually courses in their own right, and many of the interspersed "exercises" contain important results. (For Volume I a helpful solutions manual was available. Is there a similar study aid for Volume II?)

All of Professor Jacobson's books are substantial; Basic algebra is the most invitingly readable of them. It is an outcome of a lifetime's study, thought and teaching experience from a powerful and distinguished algebraist. We are indeed grateful to the author for offering us so much excellently organised material presented so clearly in this broad and deep survey of modem algebra. Bright and genuinely interested honours students, their teachers and all other mathematicians who use abstract algebra (especially those "in life's fair morning, still in the bloom of youth") will benefit immensely from both volumes of this outstanding and well-produced book.

17.1. From the Preface.

These algebras determine, by the Wedderburn Theorem, the semi-simple finite dimensional algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the Brauer-Severi varieties.

We shall be interested in these algebras which have an involution. Algebras with involution arose first in the study of the so-called "multiplication algebras of Riemann matrices". Albert undertook their study at the behest of Lefschetz. He solved the problem of determining these algebras. The problem has an algebraic part and an arithmetic part which can be solved only by determining the finite dimensional simple algebras over an algebraic number field. We are not going to consider the arithmetic part but will be interested only in the algebraic part. In Albert's classical book (1939), both parts are treated.

The largest part of our book is the fifth chapter which deals with involutorial simple algebras of finite dimension over a field.

Of particular interest are the Jordan algebras determined by these algebras with involution. Their structure is determined and two important concepts of these algebras with involution are the universal enveloping algebras and the reduced norm. Of great importance is the concept of isotopy. There are numerous applications of these concepts, some of which are quite old.

17.2. Review by: B Fein.
Mathematical Reviews MR1439248 (98a:16024).

This book presents a detailed treatment of several aspects of the theory of finite-dimensional division algebras over fields and includes a considerable amount of important material not previously available in book form. Although it assumes some familiarity with the basic theory (e.g. the Skolem-Noether theorem), it should be accessible to a reader familiar with the relevant portions of the author's standard reference [Basic algebra. II, 1989]. The author is highly selective in what he includes and the reader should be warned that some of the most important results in the theory (such as the Merkur'ev-Suslin theorem) are not mentioned in the book. For this reason, the book should be viewed as complementary to other books in this area, particularly the books by I Kersten and P K J Draxl [Skew fields, 1983]. The book has five chapters, with the last, which treats simple algebras with involution, being by far the longest.

Chapter 1 of this book is primarily concerned with skew polynomial rings and the algebras obtained from these rings by taking homomorphic images or by forming central localisations. The reduced characteristic polynomial, trace, and norm of a finite-dimensional associative algebra are also defined and used to prove the existence of separable splitting fields for central simple algebras.

Many of the main classical results of the structure theory of finite-dimensional central simple algebras over fields are obtained in Chapter 2. Both Brauer and Noether factor sets are studied and the basic properties of the Brauer group are established. The standard relations between the index and exponent of a finite-dimensional division algebra D are determined and an example of Brauer is presented which shows that these relations cannot be improved. It is shown that division algebras of index ≤4 are crossed products and an example of a non-cyclic division algebra of index 4 is given.

Chapter 3 is concerned with Brauer-Severi varieties and generic splitting fields. The author is careful to review much of the background material needed from algebraic geometry, including a discussion of Grassmannians. The main results of Amitsur and Roquette on generic splitting fields are obtained. The chapter also includes a brief treatment of the corestriction map.

The theory of central simple algebras of index a power of $p$ over a field of characteristic $p$ ($p$-algebras) is developed in Chapter 4. Albert's theorem that any $p$-algebra is similar to a cyclic algebra is proved and the Amitsur-Saltman construction of generic abelian crossed products is used to give an example of a non-cyclic division $p$-algebra.

Chapter 5 is concerned with simple algebras with involution. Let $D$ be a finite-dimensional division algebra with involution. Some highlights of this chapter include Albert's result that $D$ is a tensor product of quaternion algebras if $D$ has index 4, Rowen's result that $D$ is a crossed product for the elementary abelian group of order 8 if $D$ has index 8, Tignol's result that the algebra of 2 × 2 matrices over $D$ is a tensor product of four quaternion algebras if $D$ has index 8, and the Amitsur-Rowen-Tignol example of $D$ of index 8 which is not a tensor product of quaternion algebras. The theory of special Jordan algebras is developed and the author explains how these results play a role in the structure theory of division algebras with involution.

While the book is a valuable addition to the literature, it would have been more useful if it included an index. The book does have a substantial list of references; unfortunately, some of the items cited in the text do not appear among these references. The book, however, will be an important resource for researchers in this area and for graduate students seeking to learn more about the topics presented.