Topics in Commutative Ring Theory

Commutative algebra is one of the oldest, simplest and most beautiful branches of algebra. It developed slowly out of the convergence of three distinct lines of mathematical thought: classical geometry, number theory and the theory of equations.

It was known as early as the 18^th century that certain curves drawn in the plane can be represented by polynomial equations and that the roots of such equations gave information about the behavior of the corresponding curves. Beginning with the groundbreaking work of David Hilbert at the end of the 19^th century, commutative algebra in its modern form was finally born during the German algebraic renaissance of the 1920s, with pioneering work by the Holy Trinity of Emil Artin, B. L. van der Waerden and Emmy Noether. Its rapid growth over the next three decades was led by Oscar Zariski, Masayoshi Nagata, Claude Chevalley, Andre Weil and many others. Local field theory and algebraic multiplicity theory generalized classical invariant theory to modern algebra and completed what Hilbert began. This development reached its zenith with the completion of basic variety and scheme theory as the proper foundation for algebraic geometry by J. P Serre and Alexander Grothendieck in the late 1950s and early 1960s.

The abstraction of the new algebraic geometry had a very unfortunate effect on the teaching of commutative algebra. The new geometry was imposingly difficult, and its successes came largely due to the use of the entire machinery of commutative algebra that existed at that point. This fostered the very mistaken notion that its sole reason for existing was to provide a rigorous foundation for algebraic geometry and therefore had to be presented to students exhaustively and at the height of abstraction. Indeed, many of the classic tomes on commutative algebra are exhaustive treatises designed to give graduate students a comprehensive overview of the field in preparation for the study of the standards works in algebraic geometry. The two volume classic by Zariski and Samuels is probably the best example of this.

All this changed in the 1970s, when machines capable of real polynomial time computation became feasible. In the 1960s, the key mathematical idea had been introduced independently by Bruno Buchberger and Heisuki Hironaka. It allowed one to construct algebraic varieties and schemes from general polynomial rings using Gröbner bases. Their implementation in computer algebra systems suddenly made the subject accessible to non-experts, using only the very basics of abstract algebra. This led to a number of new applications: in computer graphics, robotics and other fields. It also had the remarkable effect of bringing the basics of algebraic geometry to the level of undergraduates, allowing students with very minimal mathematics backgrounds to focus on the pictorial aspect of the subject and the essentials of commutative algebra, which could still be mastered by anyone at this level. A number of recent books accomplished this, primarily Cox, Little and O’Shea’s soon-to-be-classic Ideals, Varieties, and Algorithms .

The above brief historical digression is important, I think, because it makes one aware of the fact that until pretty recently, a serious study of commutative algebra — one of the oldest and most beautiful topics in mathematics — was at many universities unfairly banished to the upper levels of graduate mathematics programs. The current trend is a very positive one for both subjects, as it has brought both commutative algebra and algebraic geometry down to a level for non-experts and beginning math students alike.

But, unlike 1960s style algebraic geometry, commutative algebra has never really needed such a level of difficulty; most algebraists were aware of this from the beginning. Since commutative algebra is comparatively simple, many mathematicians wanted to buck the trend and expose students to it much sooner then their second year of graduate school. The solution most came up with was to teach it selectively and purely as a branch of algebra, either ignoring or barely mentioning the connection with algebraic geometry. Symbolic of this approach are such classic books as An Introduction To Commutative Algebra, by Atiyah and MacDonald, and Irving Kaplansky’s beautiful Commutative Rings . The latter is probably most indicative of how the subject was taught 1960s and 1970s (although few could match the Chicago master’s talent for making difficult mathematics crystal clear and with his sheer depth of insight!)

Indeed, most of the basic concepts of commutative algebra are straightforward generalization of ideas most students are familiar with from high school: principal ideal domains are of course the natural generalization of the ring of integers and most of its properties extend with very little effort. As such, it’s a (forgive the pun) ideal subject to use as a first serious exposure to abstract algebra — particularly for students who aren’t specializing in pure mathematics.

Which finally brings me to John Watkin’s Topics in Commutative Ring Theory.

This slender, handsome hardcover from Princeton University Press has a nifty cover illustration of three of the subject’s founders: David Hilbert, Emmy Noether and Wolfgang Krull. This a quite appropriate front image, as this is very much a historically informed and themed introduction to the subject. The historical facts are woven into the book as sidebars in a conversational style that very much has the appearance of fleshed out lecture notes.

The book is very direct and to the point and with very little fat. Saying that might give the impression the book is relentlessly concise and dry. Nothing could be further from the truth. Watkins says in the introduction that this wonderfully written book is intended as a focused introduction to a single area of algebra for “enthusiastic” advanced undergraduates with little or no training. If the students aren’t enthusiastic about commutative rings before reading this extremely pleasant, laid back volume — they should be at the end, for the author’s own enthusiasm for this ancient subject is very apparent and hopefully will be contagious.

Chapter 1 begins — oddly enough — with a discussion of Hilbert’s solution to Gordan’s problem in invariant theory. Watkins begins here not only because he considers it the true birthplace of the subject, but also to use it as an example for beginning students of the power of the abstract viewpoint in mathematics. This approach allowed Hilbert to present a general  solution to a problem that had stumped the greatest mathematical minds for over a century. From the outset, it is clear that Watkins is writing for absolute beginners in mathematics. In this first chapter, he never precisely states the problem (a detailed presentation is given later in the book) or defines invariant theory or rings — he tells the reader who may be confused by such words “don’t worry about it right now.” Instead, he focuses on how Hilbert attacked the problem with an argument by contradiction that covers all possible cases by studying general polynomial rings. This wonderful passage is very typical of the book’s style — the path he has chosen through commutative rings is clearly a “G-rated” one, designed to show students real mathematics for the first time and to break them in slowly. Using this motivating example of how rings generalize the properties of familiar objects like integers and polynomials, he goes on in the first chapter to give all the basic definitions of rings and subrings (assuming all to be commutative with a 1, of course).

Chapter 2 defines ideals and quotient rings and their associated properties. Here you notice right away something that remains true throughout the book: for a mathematics book, the text has relatively few actual proofs of theorems. In fact, the first three chapters have virtually none, except for some in the exercises. What Watkins does  give by the ton is examples — lots and lots and lots of examples. The number of theorems which are given full proofs increases as the material’s difficulty level does — that’s to be expected in an introductory text. But Watkins, if given a choice, willoften  give several illustrative examples rather than a proof of a theorem.

This is a great choice in my opinion. Lots of books that claim to be user-friendly load up the discussion with full proofs in pedantic detail. They seem to forget that math isn’t a spectator sport and ramming lots of proofs down the students’ throats isn’t going to teach them how to devise their own. Studying a lot of well-chosen examples that are instructively presented, more often then not, will — and those the Watkins book has in abundance, all of them very clearly presented and well chosen. Frankly, I think more math texts should be written like this, especially advanced ones for serious students. If a concise text that shunted most results to the exercises included a large stock of detailed, well chosen examples alongside the definitions, I think this would make the book much more valuable for serious math majors without diluting the deliberate parsimony of words.

Chapter 3 discusses prime and maximal ideals, the ring theoretic analogues of the prime integers and simple groups, respectively. Chapter 4 gives a terrific discussion of the ring theory form of Zorn’s Lemma and a complete proof of the fact that every nonzero ring has a maximal ideal, a topic that usually isn’t discussed in books at this level and absolutely should be. There are a number of results in advanced algebra and analysis that rely heavily on the Axiom of Choice and its offspring — results that really can’t be obtained any other way. The sooner students add Zorn's Lemma to their toolbox, the better. In fact, Watkins wastes no time putting the tool to work in the next chapter on units, nilpotent elements and the nilradical, giving a full proof of the equality of the nilradical of a ring and the intersection of all its prime ideals. (I spent a very unhappy week in an advanced algebra course this semester attempting to prove it as part of a problem set. If there’s a way to do it without Zorn’s lemma, damned if I can see it. I was quite relieved when a few days after turning it in, I read the full proof in Watkins’ book.)

Chapter 6 deals with localization, a major area of ring theory and an integral part of the foundations of algebraic geometry. Watkins uses the opportunity to define and give several important examples of equivalence relations. The idea of presenting local ring embeddings as the generalization of the process of constructing the rationals from the integers is a very enlightening one. Chapter 7 gives an introduction to rings of continuous functions, a surprisingly sophisticated concept to present in a book at this level. Watkins does a good job presenting it, using just enough topology and analysis to make the definitions and examples make sense. Chapter 8 is the crux of the book, presenting a detailed account of the major ring isomorphism and homomorphism theorems. This chapter is surprisingly brief, the reason being it builds on the literally hundreds of examples given in earlier chapters and uses these results to tie them together.

Chapters 9 and 10 present the generalization of more properties of integers to three major classes of general commutative rings: unique factorization domains, Euclidean domains and principal ideal domains, using the basic properties of associates and irreducibles. It’s really in these chapters that the softness of the approach greatly restricts the presentation, as Watkins doesn’t assume any module theory or homological algebra. Chapter 11 deals with polynomial rings, using the properties developed in the previous two chapters to discuss the basic results of polynomials, such as equivalence of a unit polynomial in R[x] the polynomial having nilpotent coefficients.

Chapter 12 discusses power series rings and their more esoteric algebraic properties. Chapter 13 treats Noetherian rings, emphasizing the various forms of the Hilbert basis theorem mentioned in the first chapter and the role of the ascending chain condition in the proofs of these results. Chapter 14 gives a surprisingly coherent presentation of Krull dimension, again emphasizing concrete examples such as the Krull dimension of power series rings and discussing much of the early history of the subject. The book concludes with a long chapter on Gröbner bases; using all the tools developed in earlier chapters to give a fairly complete exposition of the basics, emphasizing their order properties.

If a single word could be used to describe this book, it would be gentle. Watkins works very hard to make sure he spells everything out clearly so that even the most inexperienced reader can follow. And yet, he doesn’t dumb the material down one bit. He does this by being incredibly selective about what to put in his presentation — module theory, category theory and homological algebra are all avoided completely. He lays the material out in the form of a grand story, rich with historical details about the subject, some of which are hard to find in any but the most exhaustive texts. For example, I was completely unaware that the first counterexample showing that it is not in general so that if a ring R is a unique factorization domain, then so is the corresponding power series ring is due to Pierre Samuel in 1961, as told at the end of chapter 12. The exercises are very good for a book at this level — not too soft, but always careful to spell out all definitions. Indeed, many of the exercises are “sidebars” — topics not defined in the text proper but defined and used in the exercises. This further extends the coverage of the book and gives students a good test of their comprehension of the basics.

I do have two little quibbles about the book: First, the book may be too gentle at times. Watkins says in the preface that he assumes advanced undergraduates are his audience with “a passing knowledge of what a groups is” — if so, why not include some module theory to enhance the presentation of UFDsand PIDs? It’s very reasonable to assume any student reading this text will have either completed or would be taking concurrently a course in linear algebra. So would it really be too much to present modules as the generalization of vector spaces?

Also, Watkins’ approach to the material is resolutely non-geometric — there are literally no pictures in the book. Surely a number of concrete pictures in the plane of affine varieties a la Cox, etc. would have assisted the already wonderfully clear presentation of polynomial rings and would have been even more effective in the chapter on Krull dimension? This really reveals a strong prejudice in Watkins towards a purely algebraic presentation — not that this is bad, that os his taste and leanings. (The references at the end prominently mention Kaplansky’s classic Commutative Rings as Watkins’ primary influence as a student. — in many ways, his book is a softer, gentler version of that classic.) But it just seems a little odd to me. In a book so clearly designed to educate students, why would you exclude such a powerful intuition-building tool?

All in all, I enjoyed the book immensely and learned a great deal from it. To me, the perfect students for this text are undergraduates just coming out of a good linear algebra text and eager to move on to a strong abstract algebra course. In fact, this book would make wonderful reading in the summer months before that algebra course. Honors freshmen could also use it for a terrific reading course. I also think the sheer historical scope of ideas makes it good collateral reading for graduate students learning commutative algebra — indeed, a terrific non-geometrically-slanted graduate level course could begin with this book and then move onto Matsumura’s Commutative Ring Theory. Watkins is to be very much commended for his effort in producing a comprehensive yet very friendly text that will greatly assist in teaching and learning this most beautiful subject.

Andrew Locascio is currently pulling his remaining hair out as a first year graduate student at Queens College of the City University of New York. Due to his chronic health problems, he has come to realize the brutal difficulty of such a career path without coffee. He has also realized his understanding of abstract algebra is nowhere near as good as he thought it was and is attempting to rectify this. He has also rediscovered his love for philosophy under the tutelage of the legendary Saul Kripke by attending his lectures on the philosophy of mathematics at the City University of New York Graduate Center. His goal by next year is to learn enough topology and algebra to be able to figure out what Dennis Sullivan is talking about.