Noncommutative noetherian rings are presently the subject of very active research. Recently the theory has attracted particular interest due to its applications in related areas, especially the representation theories of groups and Lie algebras. We find the subject of noetherian rings an exciting one, for its own sake as well as for its applications, and our primary purpose in writing this volume was to attract more participants into the area.

This book is an introduction to the subject intended for anyone who is potentially interested, but primarily for students who are at the level which in the United States corresponds to having completed one year of graduate study. Since the topics included in an American first year graduate course vary considerably, and since those in analogous courses in other countries (e.g., third year undergraduate or M.Sc. courses in Britain) vary even more, we have attempted to minimize the actual prerequisites in terms of material, by reviewing some topics that many readers may already have in their repertoires. More importantly, we have concentrated on developing the basic tools of the subject, in order to familiarize the student with current methodology. Thus we focus on results which can be proved from a common point of view and steer away from miraculous arguments which can be used only once. In this spirit, our treatment is deliberately not encyclopedic, but is rather aimed at what we see as the major threads and key topics of current interest.

The Artin-Rees property is a condition with a long history in the theory of commutative noetherian rings (where every ideal satisfies the condition). Versions of this property have also played important roles in many verifications of the second layer condition, and they place certain restrictions on the possible structure of cliques of prime ideals. We introduce a convenient form of this property and some of its uses in this chapter, which is a continuation of Chapter 12. The reader may also treat this chapter as an appendix if desired, since the Artin-Rees property will not appear later in the text aside from a few exercises in the following chapter.

• THE ARTIN-REES PROPERTY •

Definition. An ideal I in a ring I has the right AR-property if, for every right ideal K of R, there is a positive integer n such that K ∩ In ≤ KI. The left AR-property is defined symmetrically, and I has the AR-property if it has both the right and left AR-properties.

The reader should be warned that the definition just given is the weakest of several Artin-Rees properties discussed in the literature; in particular, in most of the commutative literature one finds a definition involving a stronger condition (see the proof of Lemma 13.2).

In the study of commutative rings, one meets rings of fractions early on, first as quotient fields of integral domains but later as a general construction method. Given a commutative ring R and a subset X of R, we want to find a “larger” ring in which the elements of X become units. First of all, since all products of elements of X would necessarily become units in the new ring, we may enlarge X and assume that X is multiplicatively closed and that 1 ∈ X. We then build a new ring RX−1 (often written RX in the commutative literature, but this notation can cause confusion later) as a set of fractions r/x, where r ∈ R and x ∈ X. There must be an equivalence relation on these fractions, and the situation is made slightly more complex by the fact that X may contain zero-divisors, in which case the map R → RX−1 taking r to r/1 is not injective. The correct equivalence relation turns out to be the following: We say that r/x and r′/x′ define the same element of RX−1 if and only if (rx′ − r′x)y = 0 for some y ∈ X. Some easy calculations show that we can define a ring RX−1 in this way.