One of the first constructions that an undergraduate student of algebra meets is the quotient field of a commutative integral domain, constructed as a set of fractions, that is, expressions a/b subject to an obvious equivalence relation. This leads to a very useful technique in commutative ring theory, namely, to pass from an arbitrary commutative ring R to a prime factor ring R/P and then to the quotient field of R/P. In the noncommutative case, we can ask whether it is possible to pass from a domain to a division ring built from fractions. While this is not always possible, it will turn out to be the case for any noetherian domain. However, since noncommutative noetherian rings need not have any factor rings that are domains, this is rather restrictive. Instead, recalling that prime rings are the most useful noncommutative analog of domains, we look for prime rings from which simple artinian rings can be built using fractions. The main result is Goldie's Theorem, which implies in particular that any prime noetherian ring has a simple artinian ring of fractions. It turns out to be little extra work to investigate rings from which semisimple rings of fractions can be built.

Our first task is to see how a ring of fractions can be constructed, given an appropriate set X of elements in a ring R to be used as denominators.

Since much of the current interest in noncommutative noetherian rings stems from applications of the general theory to several specific types, we present here a very sketchy introduction to some major areas of application: polynomial identity rings, group algebras, rings of differential operators, enveloping algebras, and quantum groups. Each of these areas has a very extensive theory of its own, far too voluminous to be incorporated into a book of this size. (See for instance Rowen [1980], Passman [1985], McConnell-Robson [2001], and Brown-Goodearl [2002]). Instead, we shall concentrate on surrogates – some classes of rings that are either simple prototypes or analogs of the major types just mentioned – which we can investigate by relatively direct methods while still exhibiting the flavor of the areas they represent. These surrogates are module-finite algebras over commutative rings (for polynomial identity rings), skew-Laurent rings (for group algebras), formal differential operator rings (for rings of differential operators and some enveloping algebras), and general skew polynomial rings (for some enveloping algebras and quantum groups). They will be introduced below and studied in greater detail in the following two chapters.

We will conclude the Prologue with a few comments about our notation and terminology.

• POLYNOMIAL IDENTITY RINGS •

Commutativity in a ring may be phrased in terms of a relation that holds identically, namely xy − yx = 0 for all choices of x and y from the ring.

In this chapter we study another very “classical” topic, namely, transcendental division algebras (that is, division rings which are not algebraic over their centers). While at first glance it may not appear that the general theory of noetherian rings has anything to say about division rings, we shall see that much concrete information can be gained by applying noetherian methods to polynomial rings over division rings, in particular by applying what we have learned in previous chapters about injective modules, Ore localizations, and Krull dimension. We shall, for instance, derive analogs of the Hilbert Nullstellensatz for polynomial rings over division rings and over fully bounded rings. Information about a division ring D with center k will then be obtained by developing connections among the transcendence degree of D over k, the question of primitivity of a polynomial ring D[x1, …, xn], and the Krull dimension of D ⊗k k(x1, …, xn), as well as connections between the noetherian condition on D ⊗k D and the question of finite generation of subfields of D. For technical reasons, and in order to be able to apply some of these results to Goldie quotient rings, we actually derive most of the results in this chapter for simple artinian rings rather than for division rings.

The main theme of this chapter is the exploration of the ideal theory of noetherian rings satisfying the second layer condition. This is a very large class of rings (as we began to see in the previous chapters), including many iterated differential operator rings, iterated skew-Laurent extensions, and quantized coordinate rings, as well as the group rings of polycyclic-by-finite groups and the enveloping algebras of finite dimensional solvable Lie algebras. It turns out that these rings have many properties that are not shared by other noetherian rings and that can be thought of as generalizations of well-known properties of commutative rings. We begin with a symmetry property of bimodules over these rings. This will give us immediate information about the graphs of links of these rings and will also give us the key tool to prove two intersection theorems – a strong form of Jacobson's Conjecture and an analogue of the Krull Intersection Theorem. Rings satisfying the second layer condition also behave well with respect to finite extensions. If R is a noetherian ring satisfying the second layer condition, and R is a subring of a ring S such that S is finitely generated as both a left and a right R-module, we prove that S also satisfies the second layer condition, and that “Lying Over” holds for the prime ideals in this setup.

Robert Breckenridge Warfield, Jr. (1940–1989)

Since the publication of the first edition in 1989, this book has been used by several generations of graduate students. From the accumulated comments, it became clear that a number of changes in the presentation of the material would make the book more accessible, particularly to students reading the text on their own. During this same period, the explosive growth of the area of quantum groups provided a large new crop of noetherian rings to be analyzed, and thus gave major impetus to research in noetherian ring theory. While a general development of the theory of quantum groups would not fit into a book of the present scope, many of the basic types of quantum groups are ideally suited as examples on which the concepts and tools developed in the text can be tested. Finally, readers of the first edition found a substantial list of typographical and other minor errors. This revised edition is designed to address all these points. Undoubtedly, however, the retyping of the text in TeX has introduced a new supply of typos for readers' entertainment.

Here is more detail:

Changes to the order and emphasis of topics were based, as mentioned, on the combined experience and comments of numerous students and professors who used the first edition over the past 14 years. In particular, more examples and additional manipulations with specific rings – especially in the early part of the book – were requested.

We continue the study of skew polynomial rings in this chapter, first discussing the case where the multiplication is twisted by a derivation, and then developing the general case. Since our main motivation for looking at skew polynomial rings is to be able to construct and work with further important examples of noetherian rings, most of the later part of the book could be read independently of this chapter. Readers who are interested in immediately getting into the general theory of noetherian rings should feel free to skip to Chapter 3 and return to this chapter later.

• FORMAL DIFFERENTIAL OPERATOR RINGS •

Several of the examples discussed in the Prologue appear as polynomial rings in which multiplication by the indeterminate is twisted by a derivation rather than by an automorphism. This situation has several new features – in particular, the characteristic of the ring plays an important role – but it is still significantly simpler than the general case, in which both an automorphism and a derivation act. Thus, we begin the chapter by studying the derivation case. Since many of the results in this section are parallel to ones in Chapter 1 and will be special cases of later results for general skew polynomial rings, we leave most of the proofs as exercises.

Krull dimension is a measurement of size of a ring that has an intrinsic importance of its own and is also a useful technical tool in the theory of noetherian rings. We have already discussed in the previous chapter the “classical” Krull dimension, which originated in commutative ring theory and is defined using the prime ideals of a ring. In the noncommutative theory, we need a notion of Krull dimension that does not depend on prime ideals, but which shares many of the important properties of the classical Krull dimension for commutative rings. For instance, we would like a notion of Krull dimension that gives some useful information even for simple rings. This is done by defining a dimension on modules rather than just on rings, which has the advantage that it in some sense replaces considerations involving two-sided ideals with considerations involving only one-sided ideals. The definition now used is due to Rentschler and Gabriel and will be defined in detail in the next section. While at first sight it appears completely unrelated to the classical definition, we shall see that it coincides with the classical Krull dimension on commutative noetherian rings and, in fact, on FBN rings.

Injective modules may be regarded as modules that are “complete” in the following algebraic sense: Any “partial” homomorphism (from a submodule of a module B) into an injective module A can be “completed” to a “full” homomorphism (from all of B) into A. Other types of completeness often entail similar extension properties. For instance: (a) If X and Y are metric spaces with X complete, then any uniformly continuous map from a dense subspace of Y to X entends to a uniformly continuous map from Y to X; (b) if Y is a normed linear space, then any bounded linear map from a linear subspace of Y to ℝ extends to a bounded linear map from Y to ℝ; and (c) if X and Y are boolean algebras with X complete, then any boolean homomorphism from a subalgebra of Y to X extends to a boolean homomorphism from Y to X.

In topological and order-theoretic contexts, incomplete objects can be investigated by enlarging them to their completions. Following this pattern, one way to study a module A is to “complete” it to an injective module, i.e., to embed A in an injective module E, called the “injective hull” of A, in some minimal fashion. The minimality is achieved by requiring E to be an “essential extension” of A, meaning that every nonzero submodule of E has nonzero intersection with A.

One major obstacle to adapting commutative noetherian ring theory to the noncommutative case in general is the lack of ideals. For example, the Weyl algebras over division rings of characteristic zero are simple noetherian domains, yet their module structure is quite complicated. Thus, to derive much structure theory similar to the commutative theory, one should work in a context where a large supply of ideals is guaranteed. One such context is introduced and investigated in this chapter. The results obtained may serve to give a sample of what is known about noetherian rings satisfying a polynomial identity (P.I.), although the methods used are very different from those of P.I. theory.

• BOUNDEDNESS •

Definition. A ring R is right bounded if every essential right ideal of R contains an ideal which is essential as a right ideal.

For instance, every commutative ring is right bounded, as is every semisimple ring (since a semisimple ring has no proper essential right ideals). On the other hand, a simple ring cannot be right bounded unless it is artinian. Note that a prime ring R is right bounded if and only if every essential right ideal of R contains a nonzero ideal (recall Exercise 5A).

Definition. A ring R is right fully bounded provided every prime factor ring of R is right bounded.

Many investigations of the structure of a right module A over a right noetherian ring R involve related modules over prime or semiprime factor rings of R. For instance, if A is finitely generated, then by using a prime series we may view A as built from a chain of subfactors each of which is a fully faithful module over a prime factor ring of R (see Proposition 3.13). Alternatively, we may relate the structure of A to the structure of the (R/N)-modules A/AN, AN/AN2, …, where N is the prime radical of R. Thus, we need a good grasp of the structure of modules over prime or semiprime noetherian rings. The fundamentals of such structure can be obtained with little extra effort for modules over prime or semiprime Goldie rings.

• MINIMAL PRIME IDEALS •

In working with the right Goldie quotient ring of a semiprime factor ring of a right noetherian ring R, say R/I, we shall often need to refer to the regular elements of R/I. It is then convenient to have a notation for the representatives of these cosets, as follows.

Definition. Let I be an ideal in a ring R. An element x ∈ R is said to be regular modulo I provided the coset x + I is a regular element of the ring R/I.

Goldie's Theorem gives a characterization of those rings which have a classical quotient ring that is semisimple and, in particular, artinian. This naturally gives rise to the question: Which rings have classical quotient rings that are artinian? While this question has a certain abstract interest of its own, its significance turns out to be much greater than one might initially suspect. Rings arising in a natural way frequently have artinian classical quotient rings, and this may be an important fact in their study. In particular, as we shall see in Chapter 14, if R is a subring of a ring S and P is a prime ideal in S, then, while P ∩ R need not be prime or semiprime, it is often possible to show that R/(P ∩ R) has an artinian classical quotient ring.

We first introduce a new notion of rank, known as “reduced rank” (different from the uniform rank introduced in Chapter 5), which is useful in many arguments involving noetherian rings, and we give two naive examples of its use. We then use reduced rank to derive necessary and sufficient conditions for a noetherian ring R to have an artinian classical quotient ring. This basic criterion is very satisfactory in some ways – for instance, it is phrased entirely in terms of properties of individual elements of R – but not in others.