Before we are in a position to make the planned applications to dimension theory of our work in Chapter 12 on transcendence degrees of finitely generated field extensions, we really need to study the theory of integral dependence in commutative rings. This can be viewed as a generalization to commutative ring theory of another topic studied in Chapter 12, namely algebraic field extensions.

Let F ⊆ K be an extension of fields (as in 12.1). Recall from 12.21 that an element λ ∈ K is algebraic over F precisely when there exists a monic polynomial g ∈ F[X] such that g(λ) = 0. Let R be a subring of the commutative ring S. We shall say that an element s ∈ S is ‘integral over R’ precisely w.hen there exists a monic polynomial f ∈ R[X] such that f(s) = 0. Our main task in this chapter is to develop this and related concepts. Some of our results, such as 13.22 below which shows that {s ∈ S: s is integral over R} is a subring of S which contains R, will represent generalizations to commutative rings of results in Chapter 12 about field extensions, but these generalizations tend to be harder to prove. (The more straightforward arguments which work for field extensions were included in Chapter 12, as it was thought possible that some readers might only be interested in that situation.)

Indeed, early in the development of the theory of integral dependence, we shall reach a situation where it is desirable to use a variant of what Matsumura calls the ‘determinant trick’ (see [13, Theorem 2.1]).

At the beginning of Chapter 2 the comment was made that some experienced readers will have found it amazing that a whole first chapter of this book contained no mention of the concept of ideal in a commutative ring. The same experienced readers will have found it equally amazing that there has been no discussion prior to this point in the book of the concept of module over a commutative ring. Experience has indeed shown that the study of the modules over a commutative ring R can provide a great deal of information about R itself. Perhaps one reason for the value of the concept of module is that it can be viewed as putting an ideal I of R and the residue class ring R/I on the same footing. Up to now we have regarded I as a substructure of R, while R/I is a factor or ‘quotient’ structure of R: in fact, both can be regarded as R-modules.

Modules are to commutative rings what vector spaces are to fields. However, because the underlying structure of the commutative ring can be considerably more complicated and unpleasant than the structure of a field, the theory of modules is much more complicated than the theory of vector spaces: to give one example, the fact that some non-zero elements of a commutative ring may not have inverses means that we cannot expect the ideas of linear independence and linear dependence to play such a significant rôle in module theory as they do in the theory of vector spaces.

It is time we became precise and introduced the formal definition of module.

In this chapter, we shall show how some of the techniques we have developed for handling modules over commutative Noetherian rings so far in this book can be put to good use to provide proofs of some ‘classical’ theorems. These theorems concern finitely generated modules over principal ideal domains, and include, as special cases, the ‘Fundamental Theorem on Abelian Groups’ and the ‘Jordan Canonical Form Theorem’ for square matrices over an algebraically closed field.

Basically, the main results of this chapter show that a finitely generated module over a principal ideal domain can be expressed as a direct sum of cyclic submodules, and that, when certain restrictions are placed on the annihilators of the cyclic summands, such decompositions as direct sums of cyclic modules have certain uniqueness properties.

It is perhaps worth reminding the reader of some basic facts about cyclic modules. Let R be a commutative ring. Recall from 6.12 that an R-module G is cyclic precisely when it can be generated by one element; then, G ≅ R/I for some ideal I of R, by 7.25. Since AnnR(R/I) = I and isomorphic R-modules have equal annihilators, it follows that a cyclic R-module is completely determined up to isomorphism by its annihilator.

Next, we give some indication of the general strategy of the proof in this chapter of the fact that each finitely generated module over a PID R can be expressed as a direct sum of cyclic R-modules. The following exercise is a good starting point.

This chapter is concerned more with useful techniques rather than with results which are interesting in their own right. Enthusiasm to reach exciting theorems has meant that certain important technical matters which are desirable for the efficient further development of the subject have been postponed. For example, we met Nakayama's Lemma in 8.24, but we did not develop in Chapter 8 the useful applications of Nakayama's Lemma to finitely generated modules over quasi-local rings. Also, although we undertook a thorough study of rings of fractions and their ideals in Chapter 5, we have still not developed the natural extension of that theory to modules of fractions. Again, the theory of primary decomposition of ideals discussed in Chapter 4 has an extension to modules which is related to the important concept of Associated prime ideal of a module over a commutative Noetherian ring.

These topics will be dealt with in this chapter. In addition, the theory of modules of fractions leads on to the important idea of the support of a module, and this is another topic which we shall explore in this chapter.

We begin with some consequences of Nakayama's Lemma.

9.1 Remarks. Let M be a module over the commutative ring R. By a minimal generating set for M we shall mean a subset, say Δ, of M such that Δ generates M but ℕ0 proper subset of Δ generates M.

In Chapter 14, we studied the highly satisfactory dimension theory for finitely generated commutative algebras over fields. Of course, finitely generated commutative algebras over fields form a subclass of the class of commutative Noetherian rings: in this chapter, we are going to study heights of prime ideals in a general commutative Noetherian ring R, and the dimension theory of such a ring.

The starting point will be KrulPs Principal Ideal Theorem: this states that, if a ∈ R is a non-unit of R and P ∈ Spec(R) is a minimal prime ideal of the principal ideal aR (see 8.17), then ht P ≥ 1. From this, we are able to go on to prove the Generalized Principal Ideal Theorem, which shows that, if I is a proper ideal of R which can be generated by n elements, then ht P ≥ n for every minimal prime ideal P of I. A consequence is that each Q ∈ Spec(R) has finite height, because Q is, of course, a minimal prime ideal of itself and every ideal of R is finitely generated!

There are consequences for local rings: if (R, M) is a local ring (recall from 8.26 that, in our terminology, a local ring is a commutative Noetherian ring which has exactly one maximal ideal), then dim R = ht M by 14.18(iv), and so R has finite dimension. In fact, we shall see that dim R is the least integer i ∈ ℕ0 for which there exists an M-primary ideal that can be generated by i elements.

Why write another introductory book on commutative algebra? As there are so many good books already available on the subject, that seems to be a very pertinent question.

This book has been written to try to persuade more young people to study commutative algebra by providing ‘stepping stones’ to help them into the subject. Many of the existing books on commutative algebra, such as M. F. Atiyah's and I. G. Macdonald's [1] and H. Matsumura's [13], require a level of experience and sophistication on the part of the reader which is rather beyond what is achieved nowadays in a mathematics undergraduate degree course at some British universities. This is sad, for students often find some undergraduate topics in ring theory, such as unique factorization in Euclidean domains, attractive, but this undergraduate study does leave something of a gap which needs to be bridged before the student can approach the established books on commutative algebra with confidence. This is an attempt to help to bridge that gap.

For definiteness, I have assumed that the reader's knowledge of commutative ring theory is limited to the contents of the book ‘Rings and factorization’ [20] by my colleague David Sharpe. Thus the typical reader I have had in mind while writing this book would be either a final year undergraduate or first year postgraduate student at a British university whose appetite for commutative ring theory has been whetted by a course like that provided by [20], but whose experience (apart from some basic linear algebra and vector space theory) does not reach much beyond that.

The decade since the appearance of the first edition of this book has seen the publication of some important books in commutative algebra, such as D. Eisenbud's ‘Commutative algebra with a view toward algebraic geometry’ [5], which stresses the geometric heritage of the subject, and W. Bruns' and J. Herzog's ‘Cohen–Macaulay rings’ [2], There is therefore even more motivation to encourage young people to study commutative algebra, and so, in my opinion, the raison d'être for this book – to provide ‘stepping stones’ to help young people into the subject so that they can go on to study more advanced books with confidence – is as strong as ever.

This second edition contains two new chapters, namely Chapter 16 on ‘Regular sequences and grade’ and Chapter 17 on ‘Cohen–Macaulay rings’. These chapters are just ideal-theoretic introductions to the topics of their titles: a complete treatment of them would involve significant use of homological algebra, and that is beyond the scope of the book. Nevertheless, there are some ideal-theoretic aspects which can be developed very satisfactorily within the framework of the book, and, indeed, which provide good applications of ideas developed in earlier chapters; it is those aspects which receive attention in these new chapters. It is hoped that they will whet the reader's appetite to explore Bruns' and Herzog's [2], a book which provides ample evidence of the importance of the Cohen–Macaulay condition.

I have taken the opportunity to make a few improvements to, and correct a small number of misprints in, the fifteen chapters which formed the first edition.

This is just a short chapter, the aim of which is to indicate to the reader how the direct-sum decomposition theorems of Chapter 10 can be used to derive some basic results about canonical forms for square matrices over fields. It is not the intention to provide here an exhaustive account of the theory of canonical forms, because this is not intended to be a book about linear algebra; but the reader might like to see how the ideas of this book, and in particular those of Chapter 10, can be brought to bear on the theory of canonical forms. None of the material in this chapter is needed in the remainder of the book, so that a reader whose interests are in other areas of commutative algebra can omit this chapter.

Square matrices with entries in a field K are intimately related with endomorphisms of finite-dimensional vector spaces over K.

11.1 Definitions and Remarks. Let M be a module over the commutative ring R. An R-endomorphism of M, or simply an endomorphism of M, is just an R-homomorphism from M to itself. We denote by EndR(M) the set of all R-endomorphisms of M. It is routine to check that EndR(M) is a ring under the addition defined in 6.27 and ‘multiplication’ given by composition of mappings: the identity element of this ring is IdM, the identity mapping of M onto itself, while the zero element of EndR(M) is the zero homomorphism 0: M → M defined in 6.27.

The reader should be able to construct easy examples from vector space theory which show that, in general, the ring EndR(M) need not be commutative.

Much of the remainder of this book will be concerned with the dimension theory of commutative Noetherian rings. This theory gives some measure of 'size’ to such a ring: the intuitive feeling that the ring K[X1, …, Xn] of polynomials over a field K in the n indeterminates X1, …, Xn has, in some sense, ‘size’ n fits nicely into the dimension theory. However, to make a thorough study of the dimension theory of an integral domain R which is a finitely generated algebra (see 8.9) over a field K, it is desirable to understand the idea of the ‘transcendence degree’ over K of the quotient field L of R: roughly, this transcendence degree is the largest integer i ∈ ℕ0 such that there exist i elements of L which are algebraically independent over K, and it turns out to give an appropriate measure of the dimension of R. Accordingly, in this chapter, we are going to develop the necessary background material on transcendence degrees of field extensions.

Thus part of this chapter will be devoted to the development of fieldtheoretic tools which will be used later in the book. However, there is another aspect to this chapter, as its title possibly indicates: in fact, some of the ideas of Chapter 3 about prime ideals and maximal ideals, together with results like 5.10 and 5.15 concerned with the existence of algebra homomorphisms in situations involving rings of fractions, are very good tools with which to approach some of the basic theory of field extensions, and in the first part of this chapter we shall take such an approach.