# Irving Kaplansky's books

We list below the books which Irving Kaplansky authored or co-authored. We give a selection of information such as the Publisher's description, Contents, extracts from the Preface and extracts from some reviews.

**1. Infinite Abelian Groups (1st ed. 1954, 2nd ed. 1969), by Irving Kaplansky.**

**1.1. From the Publisher.**

In the Introduction to this concise monograph, the author states his two main goals: first, "to make the theory of infinite abelian groups available in a convenient form to the mathematical public; second, to help students acquire some of the techniques used in modern infinite algebra." Suitable for advanced undergraduates and graduate students in mathematics, the text requires no extensive background beyond the rudiments of group theory.

Starting with examples of abelian groups, the treatment explores torsion groups, Zorn's lemma, divisible groups, pure subgroups, groups of bounded order, and direct sums of cyclic groups. Subsequent chapters examine Ulm's theorem, modules and linear transformations, Banach spaces, valuation rings, torsion-free and complete modules, algebraic compactness, characteristic submodules, and the ring of endomorphisms. Many exercises appear throughout the book, along with a guide to the literature and a detailed bibliography.

**1.2. Review by: Reinhold Baer.**

*Bull. Amer. Math. Soc.*

**61**(1) (1955), 88-89.

This book is written with admirable simplicity and lucidity; and it is a pleasure to be led by the author through this field, formerly so inaccessible, now so easy of access. Here it should be noted that the interesting and deep results presented in this book had been scattered through the mathematical periodicals of the world if they had been published at all. But let the reader not be misled by the apparent slimness of this monograph: There are a hundred theorems given without proof [under the misleading name of exercise] and if their proofs had been supplied [instead of the "hints" given in difficult situations] the size of the book would have been doubled at the least.

A substantial bibliography with a useful "guide to the literature" - that might be considered as a rather concentrated report on the state of the theory of abelian groups - deserves our particular attention.

We have explained why we believe that the student of the theory of abelian groups will be grateful to the author for this work. But there are various reasons why this book will be a great help to those of us who attempt to educate mathematicians and to introduce young people to the present-day ways of mathematical thinking. For instance, one likes to impress on a student's mind the central importance of structure theorems. But there don't exist many of them; and without examples of structure theorems it is impossible to explain their significance and in particular what constitutes a satisfactory structure theorem. Now these theorems form, as we mentioned before, the central theme of the book under review. Next the transfinite tools, so helpful in various branches of mathematics, are used here extensively so that the reader will learn how to avail himself of these methods. As a matter of fact this was one of the author's aims in writing this book; and in this as in his other aims he has succeeded brilliantly.

**1.3. Review by: L S Goddard.**

*Nature*

**176**(4482) (1955), 573.

A remarkable amount of material has been compressed into this monograph. After some examples of Abelian groups, definitions follow, together with some properties, of torsion groups and divisible groups. Then pure subgroups (

*Servanzuntergruppe*) and groups of bounded order are studied, and the important notion of height is introduced. This leads up to Ulm's theorem, which, involving both cardinal and ordinal numbers, gives a complete classification of countable torsion groups. The next part is concerned with the extension of preceding results to modules over principal ideal rings and an application to the theory of linear transformations. Then the ring is restricted to a discrete valuation ring R, and a study is made of a complete R-module (that is, complete in its p-adic topology). A partial classification of the submodules of a given module is made in terms of the characteristic submodules, and the account ends with a review of recent literature.

The material presented is part of a course of lectures given in 1950 in the University of Chicago. Prof I Kaplansky has a crisp, vivid style, and a pleasant feature is the interruption of the text at various points to ask a pertinent question such as "What is our goal in studying Abelian groups?", or "How do we know when we have a satisfactory theorem?". The general atmosphere of the book is one of freshness and vigour, and the reader has the satisfaction of knowing that he is being led right up to the edge of our knowledge of a branch of modern infinite algebra.

**2. An Introduction to Differential Algebra (1st ed. 1957, 2nd ed. 1976), by Irving Kaplansky.**

**2.1. From the Preface.**

Differential algebra is easily described: it is (99 per cent or more) the work of Ritt and Kolchin. I have written this little book to make the subject more easily accessible to the mathematical community. Ritt was at heart an analyst; but the subject is algebra. As a result he wrote in a style that often makes the road rough for both analysts and algebraists. Kolchin's basic paper on the Picard-Vessiot theory is admirably clear and elegant. However it isn't entirely self-contained. In particular, there is a crucial reference to an earlier paper, which in turn makes use of the Ritt theory. Certain needed facts from algebraic geometry are also likely to be troublesome to the average reader. I have sought to make the exposition as self-contained and elementary as possible. In addition to standard algebra (say the contents of Birkhoff and Mac Lane's

*Survey of Modern Algebra*), a prospective reader needs only the Hilbert basis theorem, the Hilbert Nullstellensatz, the rudiments of the theory of transcendence degrees, and a smattering of point set topology. A discerning reader will notice several places where proofs can be shortened by the use of more sophisticated techniques (Kronecker products, linear disjointness, methods from algebraic geometry).

**2.2. Contents.**

**Generalities concerning differential rings.**

Derivations.

Differential rings.

Radical ideals.

Ritt algebras.

**Extension of isomorphisms.**

Krull's theorem.

Extension of prime ideals.

A lemma on polynomial rings.

Admissible isomorphisms.

**Preliminary Galois theory.**

The differential Galois group.

The Wronskian.

Picard-Vessiot extensions.

Two special cases.

Liouville extensions.

Triangular automorphisms.

**Algebraic matrix groups and the Zariski topology.**

Z-spaces.

Ti-groups and Z-groups.

C-groups.

Solvable connected matrix groups.

A special result.

**The Galois theory.**

Three lemmas.

Normality of Picard-Vessiot extensions.

Completion of the Galois theory.

Liouville extensions.

**Equations of order two.**

The Wronskian.

Connection with a Riccati equation.

An example.

**The basis theorem and applications.**

The basis theorem.

Systems of differential equations.

The decomposition theorem.

Study of a single differential polynomial.

Examples.

**Appendix: more on matrix groups and their abstraction.**

Solvability.

CZ-groups.

Irreducible sets; the ascending chain condition.

Images of irreducible sets.

**Glossary.**

**Bibliography.**

**2.3. Review by: Hans Zassenhaus.**

*Canadian Mathematical Bulletin*

**2**(2) (1959), 134.

This is an introduction to the work of Ritt and Kolchin which is self contained and purely algebraic in character.

**2.4. Review by: Graham Higman.**

*The Mathematical Gazette*

**43**(343) (1959), 79.

This beautiful little book is precisely what its title says it is. Roughly speaking, differential algebra is what happens to the theory of commutative rings and fields when one assumes, in addition to the usual operations, one having the formal properties of differentiation. One motivation for this is provided by the desire to say exactly what is meant by the assertion that a differential equation is solvable by quadratures, and to find criteria for this to happen. The problem is evidently analogous to the problem of the solution of an algebraic equation by radicals, and since the solution of this problem is found in Galois theory, it is not surprising that Galois theory plays a central role in this book.

A problem which is by no means trivial here, though its classical analogue is trivial, is to ask what sort of group the Galois group can be. In the case dealt with in the book, the extension is obtained by adjoining the solutions of a linear equation, which form a vector space, and the elements of the Galois group are linear transformations of this space. The answer to the question turns out to be that the Galois group is an algebraic matrix group; and a particularly valuable feature of the book is the discussion of algebraic matrix groups which this necessitates. This discussion is carried further than is strictly required, and I was delighted to find at the end of it a very slick proof of the theorem that a group whose centre is of finite index has a finite derived group.

The whole book is written in the lively and immaculate style one expects from Kaplansky, and is fairly free of misprints. The use of capital letter O rather than figure 0 as the characteristic of a field is so regular that I am bound to assume it done on purpose; I hope it does not become a habit. Altogether this book should be compulsory reading for algebraists; even analysts might get something out of it.

**3. Introduction to Galois Theory (in Portuguese) (1958), by Irving Kaplansky.**

**3.1. Note.**

No information found.

**4. Some Aspects of Analysis and Probability (1958), by I Kaplansky, M Hall, Jr., E Hewitt and R Fortet.**

**4.1. Review by: Donald A Darling.**

*Journal of the American Statistical Association*

**55**(292) (1960), 786-787.

This book, as with the others in the same series, attempts to summarize the progress and present status of the subject matter treated. The exposition is as simple and elementary as is consistent with the concise and inclusive expositions the authors attempt. Even so, there are perhaps very few readers, not including the reviewer, who can intelligibly read the entire volume, and the publisher might have been well advised to publish three or four separate memoirs with a more leisurely pace. Indeed several authors attributed their brevity to lack of space.

**4.2. Review by: Hazleton Mirkil.**

*The American Mathematical Monthly*

**67**(1) (1960), 93-94.

Here are four distinct survey articles bound together in one volume. Kaplansky begins his report on Functional Analysis by briefly "reviewing the progress that has been made on the problems posed by Banach." Many mathematicians will be grateful for the existence of a published account of the present state of these problems. He then gives an expert summary (in 22 pages!!) of recent work on locally convex spaces and their operators, Banach algebras, and group representations. In particular, the classification of operators is illuminated by comparison with standard equivalence relations for matrices.

**4.3. Review by: Florence Nightingale David.**

*Science Progress (1933-)*

**47**(188) (1959), 770-771.

This is Vol. IV of "Surveys in Applied Mathematics" and it takes for its field functional analysis, combinatorial analysis, harmonic analysis and probability theory. There is presumably no connecting thread between the topics and the essays are self-contained. The "Surveys in Applied Mathematics" were undertaken partly to put the contributions to mathematical theory of the Russians, Hungarians, etc., in their correct perspective and, one supposes, to indicate how the theoretical development ties in with that of mathematicians from other countries. Three of the essays in this volume succeed in doing this, the fourth does not.

The first essay by I Kaplansky on "Functional Analysis" reviews the progress made on the problems posed by S Banach in 1932 in his

*Théorie des Opérations Linéaires*, and goes on to problems of the present day in which functional analysts are currently interested. A summary of important Russian work is given and an extensive list of useful references.

**4.4. Review by: D G Kendall.**

*Nature*

**184**(4682) (1959), 214-215.

Kaplansky's article (32 pages) is the shortest in the book, but is supported by a magnificent bibliography of 113 entries, nearly half of them concerning papers by Russian authors; he gives a remarkably clear and concise account of many topics of current interest in the theories of Banach spaces, locally convex spaces and Banach algebras. ...

The publishers are to be congratulated both on the quality of the surveys included in this volume and on their decision to publish this group of four surveys together.

**5. Rings of Operators (1968), by Irving Kaplansky.**

**5.1. Note from the Publisher.**

This book is a revised edition of notes taken by S. Berberian during a course given by Irving Kaplansky at the University of Chicago in the summer of 1955, and issued the same year in mimeographed form.

**5.2. Contents.**

1. Baer rings.

2. Equivalence of idempotents.

3. Baer *-rings.

4. The axioms.

5. Orthogonal comparability.

6. The Schröder-Bernstein theorem.

7. Infinite rings.

8. Rings of Type I.

9. Additivity of equivalence in the purely infinite case.

10. The postulate of central additivity.

11. The parallelogram law.

12. General comparability.

13. The EP and SR axioms.

14. Equivalence of left and right projections.

15. Adding polar isometries.

16. Polar decomposition.

17. Strong semi-simplicity.

18. The lattice of projections.

Appendix I: The axioms of Loomis and Maeda.

Appendix II: W*-algebras and AW*-algebras.

Appendix III: Real AW*-algebras.

Appendix IV: Algebraic simplicity of the full linear group.

**5.3. Review by: Alfred W Goldie.**

*Nature*

**223**(1969), 544-545.

This book is a revision of lectures given by Irving Kaplansky at the University of Chicago more than ten years ago, based on his research into the foundations of the theory of von Neumann and related algebras expressed abstractly as operator rings. Their appearance in the present form is a tribute to the persistent interest in this field and to the work of the author. The theory is presented in an abstract form; indeed, it is very largely a

*tour de force*in the handling of idempotent elements. It has roots, however, in the work of J von Neumann on algebras of operators in Hilbert space and in continuous geometry. So, unlikely though it seems to be at first reading, the material has natural applications in theoretical physics.

**6. Fields and Rings (1st ed. 1969, 2nd ed. 1972), by Irving Kaplansky.**

**6.1. From the Publisher.**

This book combines in one volume Irving Kaplansky's lecture notes on the theory of fields, ring theory, and homological dimensions of rings and modules.

**6.2. From the Preface.**

These lecture notes combine three items previously available from Chicago's Department of Mathematics: Theory of Fields, Notes on Ring Theory, and Homological Dimension of Rings and Modules.

**I**hope the material will be useful to the mathematical community and more convenient in the new format. A number of minor changes have been made; these are described in the introductions that precede the three sections. One point should be noted: the theorems are numbered consecutively within each section.

**Since**there are no cross-references between the sections, no confusion should result. I trust the reader will not mind a lack of complete consistency, e.g., in Part II the modules re right and the mappings are placed on the right, while in Part III both are switched to the left.

**6.3. Contents.**

Preface.

*Pt. I: Fields.*

1: Field extensions.

2: Ruler and compass constructions.

3: Foundations of Galois theory.

4: Normality and stability.

5: Splitting fields.

6: Radical extensions.

7: The trace and norm theorems.

8: Finite fields.

9: Simple extensions.

10: Cubic and quartic equations.

11: Separability.

12: Miscellaneous results on radical extensions.

13: Infinite algebraic extensions.

*Pt. II: Rings.*

1: The radical.

2: Primitive rings and the density theorem.

3: Semi-simple rings.

4: The Wedderburn principal theorem.

5: Theorems of Hopkins and Levitzki.

6: Primitive rings with minimal ideals and dual vector spaces.

7: Simple rings.

*Pt. III: Homological Dimension.*

1: Dimension of modules.

2: Global dimension.

3: First theorem on change of rings.

4: Polynomial rings.

5: Second theorem on change of rings.

6: Third theorem on change of rings.

7: Localization.

8: Preliminary lemmas.

9: A regular ring has finite global dimension.

10: A local ring of finite global dimension is regular.

11: Injective modules.

12: The group of homomorphisms.

13: The vanishing of Ext.

14: Injective dimension.

Notes.

Index.

**6.4. Review by: A Rosenberg.**

*Mathematical Reviews*.

In all three parts of this book the author lives up to his reputation as a first-rate mathematical stylist. Throughout the work the clarity and precision of the presentation is not only a source of constant pleasure but will enable the neophyte to master the material here presented with dispatch and ease.

**6.5. Review by: K Blackburn.**

*The Mathematical Gazette*

**55**(393) (1971), 339-340.

Three sets of Professor Kaplansky's lecture notes, each of which has been previously available through the Mathematics Department at Chicago University, have been edited and combined into a single volume under the general title

*Fields and Rings*. More explicitly, the three sections of the book cover certain topics in Galois theory, non-commutative ring theory and the theory of homological dimension. ...

These notes represent a very readable anthology of topics from ring and field theory ranging from modern interpretations of classical theory to topics of current interest. The author thoughtfully introduces and motivates each new topic as his survey develops, occasionally digressing a little to embrace some mathematical result which takes his eye, and one senses the spontaneity inherited from the 'live' origins of the notes, even to the inclusion of the occasional shrewd, humorous comment. Anyone who has had the pleasure of hearing Professor Kaplansky speak will recognise the same easy style reflected in these notes. The notes have withstood the critical analysis of live audiences and their excellence is guaranteed by the scholarship of the author. One or two inconsistencies of notation occur between the three parts of the book, of which the author is aware, but these represent a minor hindrance. In all, the text should provide stimulating reading for anyone whose interests embrace ring theory.

**7. Linear Algebra and Geometry. A Second Course (1st ed. 1969, 2nd ed. 1974), by Irving Kaplansky.**

**7.1. From the Preface.**

Linear algebra, like motherhood, has become a sacred cow. It is taught everywhere; it is reaching down into the high schools and even the elementary schools; it is jostling calculus for the right to be taught first.

Yet all is not well. The courses and books all too often stop short just as the going is beginning to get interesting. And classical geometry, linear algebra's twin sister, is a bridesmaid whose chance of getting near the altar becomes ever more remote. Generations of mathematicians are growing up who are on the whole splendidly trained, but suddenly find that, after all, they do need to know what a projective plane is.

At the University of Chicago we have for years offered two quarters of linear algebra. In the second quarter there is time to push on to something more advanced. Whenever l have taught this course l have presented various subsets of the material in Chapters 1 and 2. We offer one quarter of geometry. At present no one is required to take it. There has been an inexorable tendency to assume that the customers have had ever more linear algebra. Chapter 3 represents my answer to the problem: given a roomful of students who have been crammed full of linear algebra, what geometry should you teach them, and how?

I have been a little arbitrary, but I hope not excessively so, in the selection of topics. I touch on similarity lightly because current books are especially strong there. I cautiously stay away from dual spaces for a while to keep the opening of the book quite elementary, but then use them enthusiastically. Inner product spaces get VIP treatment because I think they are enormously important. Topics related to convexity (Frobenius's theorems on positive matrices, Farkas's theorem of the alternative for two-person zero-sum games, etc.) are omitted entirely, because there l felt least able to make a contribution.

I realise I am not alone in trying to reunite the threads linking algebra and geometry. Let me mention in particular the splendid book of Gruenberg and Weir; l highly recommend it to readers (as a prelude to mine, of course).

I owe a great deal to a great many people. Among outstanding debts that are a pleasure to acknowledge I would like to mention the papers and books of Dieudonne and Jacobson, which taught me so much about inner product spaces, the classic of Halmos on finite-dimensional Hilbert spaces that has since been shamelessly cribbed by everyone (including me), and the geometry courses I took from H S M Coxeter at the University of Toronto. (I hope he will enjoy the enthusiasm he did so much to instil, and forgive the defects.)

**7.2. From the Publisher of the 1974 edition.**

The author of this text seeks to remedy a common failing in teaching algebra: the neglect of related instruction in geometry. Focusing on inner product spaces, orthogonal similarity, and elements of geometry, this volume is illustrated with an abundance of examples, exercises, and proofs and is suitable for both undergraduate and graduate courses. 1974 edition.

**7.3. From the Publisher of the Dover reprint (2003).**

A prominent and influential mathematician who has received numerous awards wrote this text to remedy a common failing in teaching algebra: the neglect of related instruction in geometry. Based on his many years of experience as an instructor the University of Chicago, author Irving Kaplansky presents a coherent overview of the correlation between these two branches of mathematics, illustrating his topics with an abundance of examples, exercises, and proofs. Suitable for both undergraduate and graduate courses.

Unabridged republication of the edition published by Chelsea Publishing Company, New York, 1974.

**7.4. Review by: M Hunacek.**

*Mathematical Association of America.*

https://www.maa.org/press/maa-reviews/linear-algebra-and-geometry-a-second-course

When Irving Kaplansky died in 2006, the mathematical community lost both a first-rate mathematician and an exceptional writer. His gifts for lucid, compelling exposition are very much evident in this superb book, which I read in its first edition about forty years ago and from which I first became aware of the extent to which linear algebra can be used as a tool to study (and in fact actually define) various topics in Euclidean and non-Euclidean geometry.

I had sat in on an undergraduate course taught by David Bloom which related the two subjects (and which later morphed into Bloom's textbook Linear Algebra and Geometry) but it was the book under review that taught me for the first time about the strong and beautiful connections between projective geometry and linear algebra.

Kaplansky's book, however, is not just about geometry from the linear-algebraic viewpoint; in fact, geometry does not make an appearance until the third and final chapter of the book, after some fairly sophisticated linear algebra has been discussed in the first two. The book (a Dover republication of the second edition, which as far as I can tell differs from the first only in the inclusion of a very brief Addendum mentioning a few developments that occurred between 1969 and 1974) ...

There aren't very many books that convey mathematics as well as this one, and in such an insightful and pleasant way. I thought this book was exceptional when I first read it, and, looking at it again, I see nothing to make me revise that opinion.

**7.5. Review by: Robert J Troyer.**

*The American Mathematical Monthly*

**78**(3) (1971), 314-315.

This book contains an elegant treatment of topics from linear algebra and geometry which the author has taught in a two term sequence at the University of Chicago. The first two chapters are devoted to linear algebra and the third chapter is to geometry. ... In the judgment of the reviewer, this text is best suited for honours undergraduate students or beginning graduate students. The concise exposition, the sparseness of concrete examples and the exercises would seem to limit the use of this book as a text. Nevertheless, anyone searching for a textbook on linear algebra and geometry for capable students would do well to give this book careful consideration.

**8. Commutative Rings (1st ed. 1970, 2nd ed. 1974), by Irving Kaplansky.**

**8.1. From the Preface.**

This account of commutative rings has grown over the years through various stages. The first version was an appendix to the notes on homological dimension issued in 1959. The second was a crude 49 page dittoed manuscript written in 1961. This account was expanded in the 1965-6 course presented at Queen Mary College. I owe an enormous debt to Professor Paul M Cohn of Bedford College for his expert job in writing up the course; large parts are incorporated here with only small changes. A draft of the book was prepared during the summer of 1968, and was used for a course at Chicago during 1968-9.

No attempt has been made to achieve scholarly completeness. References to sources have been made only in scattered instances where it seemed particularly desirable, and the bibliography contains only items to which there is an actual reference. My intention is to give an account of some topics in the theory of commutative rings in a way that is accessible to a reader with a modest background in modern algebra; I assume only an acquaintance with the definitions and most elementary properties of rings, ideals, and modules. More exactly, this is true till §4-1, where I presuppose the theory of homological dimension, and §4-4 where use of the long exact sequence for Ext begins. I hope that readers will find it feasible to go on from this book to a deeper study of the literature. It would be most urgent to learn the theory of completion and the Cohen structure theorems.

I have inserted numerous exercises, partly to cover additional material of lesser importance, and partly to give the reader a chance to test his growing skill. There is occasional reliance on the exercises as part of the exposition, and I hope no reader will find this inconvenient.

In the style of Landau, or Hardy and Wright, I have presented the material as an Unbroken series of theorems. I prefer this to the n-place decimal system favoured by some authors, and I have also grown tired of seeing a barrage of lemmas, propositions, corollaries, and scholia (whatever they are). I admit that this way the lowliest lemma gets elevated to the same eminence as the most awesome theorem. Also, the number of theorems becomes impressive, so impressive that I felt the need to add an index of theorems.

All rings have a unit element, except for a fleeting instant in Ex. 22 of §2-2. All rings are commutative except in several (not quite so fleeting) isolated passages.

**8.2. Review by: R A Smith.**

*The American Mathematical Monthly*

**79**(1) (1972), 99.

It is not often that a new mathematics text appears on the market which simultaneously offers the student the wealth of ideas as appears in this book and is in addition a subject so new as commutative rings. The book is written in a leisurely style which invites the reader to continue. The proofs are generally complete and indeed are works of art. The presentation follows the definition, theorem and remark style, so that "the lowliest lemma gets elevated to the same eminence as the most awesome theorem." There are a vast number of exercises, many of which contain a substantial hint. Each section of the book is well motivated, though global motivation is largely lacking. It is a pity that Kaplansky did not include some historical development of the subject, as he did, for example, in his lecture on Commutative Rings addressed to the 1968 Canadian Mathematical Congress at the University of Manitoba. ...

This book is a highly welcome addition to the existing literature on commutative rings. It can very well serve as the basis for an undergraduate or graduate level course following an introduction to modern algebra

**9. Lie Algebras and Locally Compact Groups (1st ed. 1971, 2nd ed. 1974), by Irving Kaplansky.**

**9.1. From the Publisher.**

This volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, "Lie Algebras," the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, "The Structure of Locally Compact Groups," deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952.

**9.2. Contents.**

PREFACE

*Chapter I. LIE ALGEBRAS.*

1. Definitions and examples.

2. Solvable and nilpotent algebras.

3. Semi-simple algebras.

4. Cartan subalgebras.

5. Transition to a geometric problem (characteristic 0).

6. The geometric classification.

7. Transition to a geometric problem (characteristic p).

8. Transition to a geometric problem (characteristic p), continued.

*Chapter II. THE STRUCTURE OF LOCALLY COMPACT GROUPS.*

1. NSS groups.

2. Existence of one-parameter subgroups.

3. Differentiable functions.

4. Functions constructed from a single Q.

5. Functions constructed from a sequence of Q's.

6. Proof that i/n. is bounded.

7. Existence of proper differentiable functions.

8. The vector space of one-parameter subgroups.

9. Proof that K is a neighborhood of 1.

10. Approximation by NSS groups.

11. Further developments.

BIBLIOGRAPHY.

INDEX.

**9.3. From the Preface.**

These lecture notes combine three items previously available from Chicago's Department of Mathematics: Theory of Fields. Notes on Ring Theory, and Homological Dimension of Rings and Modules. I hope the material will be useful to the mathematical community and more convenient in the new format.

A number of minor changes have been made; these are described in the introductions that precede the three sections.

One point should be noted: the theorems arc numbered consecutively within each section. Since there arc no cross-references between the sections, no confusion should result.

I trust the reader will not mind a lack of complete consistency, e.g., in Part II the modules are right and the mappings are placed on the right, while in Part III both get switched to the left.

**10. Set Theory and Metric Spaces (1st ed. 1972, 2nd ed. 1977), by Irving Kaplansky.**

**10.1. From the Preface.**

I first taught a course on set theory and metric spaces in the autumn of 1949. In subsequent years Edwin Spanier presented the material in a somewhat similar way, and he prepared an excellent set of mimeographed notes. These notes were used repeatedly as a text at Chicago.

I have now put them into a somewhat more definitive form. I am very grateful to Spanier for courteously allowing me to complete a project in which he was so deeply involved, and for permission to incorporate numerous exercises from his notes.

The two halves of the book are of nearly equal size. The set theory (with a bow to Halmos) is super-naive. Axiomatic set theory is barely mentioned. The paradoxes get some attention, but in effect they are brushed aside as not really being menacing. My intention is to present the set theory that a working mathematician really needs ninety-nine per cent of the time, and a little bit more (on the theory that to be sure of doing enough, you must do more than enough). A little knowledge may not be a dangerous thing, but a little axiomatic set theory is just not much fun; I hold that one should either take a healthy bite or leave it out in toto.

On the other hand, I hope that this book will help to fight those who say that set theory is a luxury. Hilbert vowed that no one would ever drive us out of the paradise created by Cantor. Let those who agree strive diligently to transmit to the next generation the knowledge that there is such a paradise.

In the metric space half of the book I have tried to cover the basic topics with a helpful amount of detail and motivation. I hope it will be found useful by teachers who share my belief that topology is best introduced first in the less austere setting of metric spaces. A final appendix has been added to help bridge the gap between metric and topological spaces.

**10.2. Contents.**

Basic Set Theory:

1.1. Inclusion.

1.2. Operations on sets.

1.3. Partially ordered sets and lattices.

1.4. Functions.

1.5. Relations.

Cartesian products Cardinal Numbers:

2.1. Countable sets.

2.2. Cardinal numbers.

2.3. Comparison of cardinal numbers. Zorn's lemma.

2.4. Cardinal addition.

2.5. Cardinal multiplication.

2.6. Cardinal exponentiation. Well-Ordering.

The Axiom of Choice:

3.1. Well-ordered sets.

3.2. Ordinal numbers.

3.3. The axiom of choice.

3.4. The continuum problem.

Basic Properties of Metric Spaces:

4.1. Definitions and examples.

4.2. Open sets.

4.3. Convergence. Closed sets.

4.4. Continuity.

Completeness, Separability, and Compactness:

5.1. Completeness.

5.2. Separability.

5.3. Compactness.

Additional Topics:

6.1. Product spaces.

6.2. A fixed-point theorem.

6.3. Category.

Appendixes:

1. Examples of metric spaces.

2. Set theory and algebra.

3. The transition to topological spaces Selected bibliography Index.

**10.3. Review by: Wayne R Park.**

*The American Mathematical Monthly*

**80**(8) (1973), 953-955.

Throughout the set theory presentation, the author has included fascinating personal notes about the lives and work of men in the field, Cantor through Cohen. I had to smile when Kaplansky gives a quotation from Cantor's original paper, leaving it untranslated in the German with the comment that the German "seems particularly suitable for discourse of this kind." The metric space development is very well organized. Theorems and proofs and exercises are put together in a smooth pattern with the end result that there are no overly strenuous proofs. As already mentioned, flow in reading is something we don't often see in a mathematics text. Here it is exceptionally well done. ...

What will make this text an excellent learning tool are the exercises. There are ample straightforward problems to help the reader get started and also some very challenging and fascinating "starred" problems. The elementary exercises form an essential part of the later presentations in the book. It is especially pleasing to see numerous applications of the material on cardinality in the metric space exercises.

I must say a few words of caution for the student or teacher who might use this book as an introductory text on the subject. The reader is expected to have some mathematical maturity before he attempts the text. He is expected to be well versed on the real numbers, their construction and general properties. If the student has attained a reasonable competence in proof writing from this background or other sources, he should fare quite well using this book. If not, I would expect some consternation in trying to relate the concept of exactness in proof with the apparent freeness of approach to the set theory which the author initially uses. If the teacher of this subject is prepared to handle this situation, I feel the book will be an outstanding text to use.

I do hope that members of the mathematics profession will take the opportunity to glance through this work. The value of good literary style, I feel, has not been adequately stressed in mathematical literature. Here, Kaplansky gives us a fine example of how to do it right and well - an outstanding book.

**10.4. Review by: Lawrence Corwin.**

*American Scientist*

**66**(4) (1978), 507.

The first part of this book is a treatment of basic set theory, much in the style of

*Naive Set Theory*, by Halmos (but perhaps slightly more naive). The second part deals with the theory of metric spaces. The style is bright, and there is a good selection of exercises. Although the working mathematician is unlikely to find anything new here, students may find the text a relatively painless introduction to set theory and topology. A series of appendixes present, among other things, applications of Zorn's lemma to algebra and a discussion of how metric spaces generalise to become topological spaces.

**11. Matters Mathematical (1978), by Israel N Herstein and Irving Kaplansky.**

**11.1. Contents.**

Sets and Functions.

1. Sets.

2. Sets and counting.

3. Functions.

Number Theory.

1. Prime numbers.

2. Some formal aspects.

3. Some formal consequences.

4. Some basic properties.

5. Equivalence relations.

6. Congruence.

7. Applications of the pigeon-hole principle.

8. Waring's problem.

9. Fermat and Mersenne primes.

Permutations.

1. The set A(S).

2. Cycle decomposition.

3. Even and odd.

4. The interlacing shuffle.

5. The Josephus permutation.

Bibliography.

Group Theory.

1. Definition and examples of groups.

2. Some beginning notions and results.

3. Subgroups.

4. Lagrange's theorem.

5. Isomorphism.

Finite Geometry.

1. Introduction.

2. Affine planes.

3. Counting arguments.

4. Planes of low order.

5. Coordinate affine planes.

6. Parallelograms and midpoints.

7. The nonexistence of planes of order 6.

Suggestions for further reading.

Game Theory.

1. Probability.

2. Mathematical expectation.

3. Preliminary remarks on games.

4. Examples.

5. The general two-by-two game.

6. Simplified poker.

7. The game without a name.

8. Goofspiel.

Suggestions for further reading.

Infinite Sets.

1. Infinite numbers; countable sets.

2. Uncountable sets.

Index.

**11.2. From the Preface.**

This book is based on notes prepared for a course at the University of Chicago. The course was intended for nonmajors whose mathematical training was somewhat limited.

Our aim is to cover a selection of topics that give something of the flavor of modern mathematics. Furthermore, we try to carry each topic far enough to prove something substantial . Mastery of the material requires nothing beyond the algebra and geometry normally covered in high school.

In transforming the notes into a book, we expanded the material considerably. As a result we feel that the book may be suitable for a wider audience. For instance, it could be used in courses designed for students who intend to teach mathematics. Several of the topics touch directly on matters that the

**prospective**teacher might be teaching his own students: geometry, probability and game theory, and portions of the number theory.

**Another**possible use of the book is in a course for mathematics majors as a precursor to more specialized courses. Permutations, groups, and infinite sets could serve as their introduction to abstract mathematics.

We want the reader to see mathematics as a living subject in which new results are constantly being obtained. Whenever possible, we mention the most recent advances in the area under discussion.

Number theory has fascinated mankind for centuries. The subject abounds in open questions that are easily stated and easily understood. We open our long Chapter 2 with a preliminary informal section that surveys some of these challenges. This should motivate the more careful treatment that follows.

In much of modern mathematics one studies abstract systems. This is done riot for the sake of generality alone, but because it has been found to be effective in solving classical problems. The abstractions evolved from special situations. A notable example is the transition from permutations to groups. Chapters 3 and 4 illustrate this: a "concrete" special case as a prelude to an abstraction. Permutations are easily grasped. They lend themselves to a variety of charming experiments; and nontrivial theorems and challenging open questions are not at all lacking. In this way the ground is prepared for the general concept of a group.

In Chapters 5 and 6 we treat two topics that are nearly self-contained: finite geometries and games. This is very much twentieth-century mathematics, and yet a minimum of technique is

**needed**as a prerequisite. The high spot of Chapter 5 is the Bruck-Ryser theorem; that of Chapter 6 is the theory of two-by-two games.

Many students are introduced to set theory in grade school or high school. Chapter 1 reviews this material for such students and establishes the language we use throughout the book. We felt that it would be interesting to push beyond

**this**modest level into the Cantor theory of infinite sets. People find infinity mysterious. The little that Chapter 7 contains on the cardinal equivalence of infinite sets sheds some light on these mysteries and introduces the reader to a remarkable chapter of current mathematics.

**We try to avoid manipulation and technical details**as much as possible, keeping the central ideas in the foreground. This cannot always be done. To get to the crux of a substantial piece of mathematics requires hard work and technique. But we believe that the book is within reach of people with the background we mentioned at the outset.

**11.3. Review by: Allen Stenger.**

Mathematical Association of America.

https://www.maa.org/publications/maa-reviews/matters-mathematical

This is a survey of several areas of mathematics, chosen because they of representative of current research and because they can be understood without a lot of mathematical background. The authors say in the Preface, "We want the reader to see mathematics as a living subject in which new results are constantly being obtained." The title is a quote from the Major-General's Song in Gilbert & Sullivan's The Pirates of Penzance and indicates the book's broad and varied scope.

The book often recurs to earlier topics and is not as miscellaneous as it looks on first glance. For example, it starts with elementary set theory but comes back at the end of the book to look at transfinite cardinals. (And it's refreshing to see a discussion elementary set theory that does not begin and end with Venn diagrams.) Similarly there is a chapter on permutations that is followed by a chapter on group theory.

The book is now 35 years old but has aged well, and the subjects covered are still lively research topics even though the statuses given here are not completely up to date. The big weakness of the book, in my view, is that it concentrates on mathematical concepts rather than mathematical problems. Giving more attention to problems would have made it more interesting and would have reinforced the message that math is a living subject.

This is an unusual book and it doesn't fit nearly into a well-known category or have an obvious audience. Although its prerequisites are low, it is a rigorous math book, with definitions, proofs, abstraction, some intricate reasoning, and challenging exercises. The authors suggest that "it could be used in courses designed for students who intend to teach mathematics". I think it would be most useful for pre-service or in-service high-school math teachers, especially those who want a good grounding in mathematical processes. It's definitely not a "popular math" book, and is probably too hard for a college math appreciation class.

**11.4. Review by: Johnston Anderson.**

*The Mathematical Gazette*

**64**(427) (1980), 61-62.

Let me begin by praising this book. The authors are distinguished, indeed, in at least one case, quite famous; the printing is excellent and the misprints and errors seem to be few; the diagrams are clear and the binding is sound.

Unfortunately, these virtues hardly compensate for the contents. I have now read this book three times. After the first perusal, I found myself so incensed that I put the book aside and did not even attempt a review, feeling that I should wait until my reactions had cooled and hoping that maybe the book would improve with age, like port. Instead, it rather grows on one with the passage of time, like mildew.

To quote the authors, the original target audience were "non-specialists whose mathematical training is somewhat limited" but, with the growth of the text from lecture notes to book, the authors feel confident to include within its ambit would-be teachers of mathematics and even those who will go on to specialise in mathematics. The contents, also according to the preface, "aim to cover a selection of the topics that give something of the flavour of

*modern mathematics*" (my italics). ...

My disillusionment, frustration and irritation is not all philosophical, however; it extends to the detailed execution of the text as well. There is a spurious attempt to make the poor benighted non-specialist feel at home: little biographical snippets printed in a different font and a different size, just sufficient to divert the reader from his train of thought, smatterings of history that are not really historical in the proper sense but catalogues of dates and events (who on earth wants to know this without being aware of the historical

*mathematical*background?), efforts at breathless immediacy ("in 1973, Hagis and McDaniel showed that an odd perfect number must be larger than $10^{50}$ and have a factor of at least 11213"), examples on sets (American presidents, baseball pitchers, readers of

*Harper's*and

*Playboy*), which not only twee, but culture-dependent, and also the kind of thing that students find both unrealistic and patronising. Yet despite these efforts the underlying format is still the same boring old Definition-Theorem-Proof that we know so well and which even specialist mathematicians find difficult. Examples usually illustrate rather than motivate - in a book so leaning towards abstraction, one would have thought that motivation through examples for concepts and definitions might have been more useful. Furthermore, the treatment of both formal and informal parts of the text is uneven; sometimes, laborious detail is used to demonstrate some fairly elementary point while, at other times, the terse, concept-dense sentences of a typical textbook proof will be deployed without compunction.

If one may borrow the wonderfully pungent language of J M Hammersley, writing six years before the first edition of this book, this is yet another episode in the enfeeblement of traditional mathematical skills by modern mathematics and similar soft intellectual trash geared to the imagined needs of liberal-arts-type students.

**11.5. Review by: Stuart J Sidney.**

*American Scientist*

**67**(2) (1979), 247.

Based on a course given at the University of Chicago, this is a cross between a text and an appreciation. As a text it offers a systematic exposition for the non specialist of the elementary parts of several "noncontinuous" areas in the mainstream of modern mathematics, exercises included. As an appreciation, it offers "snapshots" of some of the deeper and more striking applications of the general theory (some of which may be unfamiliar to the average professional), a good deal of mathematical history, and a fine sense of mathematics as a continuously developing body of knowledge.

The chapter headings are: sets and functions, number theory, permutations, group theory, finite geometry, game theory, infinite sets. Since many other concepts appear in passing - for example, rings and probability - there is material on most key ideas which involve neither linear algebra nor topology (and limits). This is a good place for a general audience to get an idea of what mathematics is about today.

**11.6. Review by: Kenneth C Skeen.**

*The Mathematics Teacher*

**68**(5) (1975), 409.

Here's a nicely done book for an appropriate level of study. Chapter topics include sets and functions, number theory, permutations, group theory, finite geometry, game theory, and infinite sets.

The authors suggest it for "nonmajors whose mathematical training was somewhat limited." Later, after expanding the original course, they add "a wider audience," including prospective teachers.

Topics and methods of presentation seem much more appropriate for the latter audience, who, one might anticipate, would enter the course with two or three years of study in a mathematical major. For these, it would seem highly interesting.

Last Updated July 2020