Posted by Jason Polak on 31. January 2014 · Write a comment · Categories: commutative-algebra · Tags: ,

P1000536It was more than a year ago that I opened a package that I got in the mail, taking out this green ex-library hardcover in excellent condition. Now, I honestly can’t remember what prompted me to order it (perhaps it was the author’s name), but I remember reading the first few sections and feeling that it would be worthwhile to spend some time over its pages to learn more about those primes. Unfortunately, I didn’t manage to keep reading at the time, but a few months ago I decided to push through this volume with a little spare time I had, and this became my first real serious conversation with the zero divisor.

Certainly, no algebraist can ever escape the grasp of the zero divisor. In a ring $ R$, a nonzero element $ r\in R$ is called a zero divisor if $ rs = 0$ for some nonzero $ s\in R$. In even basic questions on ring theory, zero divisors are bound to be lurking. Our topic today is commutative rings, so we’ll assume from now on that $ R$ is commutative. The rings probably easiest to understand, at least if we’re not considering relations to other rings, are fields. If $ R$ is not a field, then it has an ideal $ I$ that is not prime so $ R/I$ already has zero divisors. So even if $ R$ is a domain, some of its quotients will not be as long as $ R$ is not a field.

Consider the set $ S\subseteq R$ of non-zerodivisors. Clearly, $ S$ is multiplicatively closed: in other words, if $ x,y\in S$ then $ xy\in S$. Moreover, if $ z$ divides $ x$ then $ z$ is not a zero divisor either. We then call $ S$ saturated. Now, any saturated multiplicatively closed set is the set-theoretic complement of a union of prime ideals, so the set of zero divisors in $ R$ is the union of of prime ideals.

We can consider a more general concept: if $ M$ is an $ R$-module then we say that $ r\in R$ is a zero divisor on $ M$ if $ rm = 0$ for some $ m\in A$. Let us use the notaiton $ Z(M)$ for the zero-divisors on $ M$. It turns out that when $ R$ is Noetherian and $ M$ is finitely generated, we can write $ Z(M)$ as a finite union of primes maximal in $ Z(M)$, each being the annihilator of a nonzero element in $ M$. This theorem, not difficult to prove, is one of the most valuable observations in the theory of Noetherian rings.

It is in Irving Kaplansky’s book “Commutative Rings” that variations on the theme of zero divisors are drawn out into various stories, starting with the study of prime ideals in Chapter 1. It then proceeds to a deeper study of Noetherian rings, zero divisors, and integrality in Chapter 2, culminating in a slightly technical section on the intersection of local domains.

Chapter 3 contains what is perhaps the most important serious definition in the book: that of a regular sequence. It is the theme of the regular sequence where zero divisors and non-zerodivisors dance together on the precipice of that slash designating the quotient ring.

A regular sequence on an $ R$-module $ M$ is a sequence $ x_1,\dots,x_n$ of elements in $ R$ such that $ x_1$ is not a zero divisor on $ M$ and in general $ x_i$ is not a zero divisor on $ M/(x_1,\dots,x_{i-1})M$; moreover, we also require $ M\not= (x_1,\dots,x_n)M$. Why are these regular sequences important? If $ I$ is an ideal in $ R$ and $ R$ is Noetherian, then every maximal regular sequence in $ I$ on an $ R$-module $ M$ has the same length and this is called the grade of $ I$ in $ M$, denoted by $ G(I,M)$. If $ M = R$ we just call this number the grade of $ I$. This gives us an invariant of $ I$, and it is related to the rank of $ I$ simply in that it is always less than or equal to the rank. (Note, the rank of an ideal that is not prime is defined as the infimum over the ranks of all primes containing that ideal.)

Rings for which the grade and rank coincide for every ideal are called Macaulay, and examples include among others polynomial rings over Macaulay rings, such as fields. The property of Macaulay, and the geometric significance is expounded upon in Eisenbud’s book (Chapters 17-18). The geometric subtleties are nowhere to be found “Commutative Rings”, which is definitely a book purely about rings. This is hardly an omission however, and probably would have hurt the natural flow of this book. Another notable contribution of this third chapter is the proof of Krull’s principal ideal theorem: if $ P$ is a prime minimal over a proper principal ideal $ (x)$ then $ P$ has rank at most one. An amusing pair of exercises is provided to prove that $ P$ has rank one if $ x$ is a non-zerodivisor and it has rank zero if $ x$ is nilpotent—but I digress. Krull’s theorem is then generalised and used to prove that $ R/x$ being Macaulay implies $ R$ Macaulay if $ x$ is in the Jacobson radical of $ R$, a result that can be used with lethal effectiveness in inductive arguments.

The final Chapter 4 closes with some homological techniques, and contains a major theorem proved homological style: a regular local ring is a unique factorisation domain. As warned earlier in the book, the proofs towards the end become a bit terse, but certainly understandable if one is willing to use a little more paper and pencil. This chapter also assumes the change of rings theorem for projective dimension but proves the dual notions for injective dimension. It is unfortunate that this last chapter did not continue by exploiting the Koszul complex associated to a regular sequence, although the interested reader can consult the appropriate chapters in Eisenbud. In fact, virtually everything in this book is in Eisenbud’s book “Commutative Algebra”, albeit with much more detail, so that the interested reader can pursue some topics more deeply with Eisenbud after “Commutative Rings”.

Throughout this book, the exposition is lucid and makes much sense, even on the grand scale of the book. Kaplansky manages to avoid being overly Bourbaki-like by inserting ample comments and discussion that are pulsing with personality. Moreover, it is clear that “Commutative Rings” was written by an algebraist intimately connected with the material, presenting a refreshing contrast to the awkward style present in the dusty corners of algebraic geometry texts everywhere.

Succinctly, Kaplansky’s little volume is a book that will, in 180 pages and 266 solvable exercises, refine and sharpen the intuition of someone already familiar with the basics of commutative rings, their modules, and homological algebra. “Commutative Rings” will surely be of great value to any student in algebra, algebraic geometry, and related fields.

Recommended Prerequisites: A basic familiarity with rings and modules is assumed. In Chapter 4, some basic homological algebra such as the $ \mathrm{Ext}$-functor is also assumed, and some exposure to homological dimension theory would be very helpful. The first four chapters of Weibel’s book are more than sufficient. I would expect that this book would appeal mostly to people who already have an ample supply of commutative rings and modules in their mind. Category theory is not used in this book.

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