NOETHERIAN RINGS and their applications

http://dx.doi.org/10.1090/surv/024

= 1 1 MATHEMATICAL SURVEYS It ,11 AND MONOGRAPHS

fr==iJll NUMBER 24

NOETHERIAN RINGS and their applications

LANCE W. SMALL, EDITOR

American Mathematical Society Providence, Rhode Island

1980 Mathematics Subject Classification (1985 Revision). Primary 16A33; Secondary 17B35, 16A27.

Library of Congress Cataloging-in-Publication Data

Noetherian rings and their applications. (Mathematical surveys and monographs, 0076-5376; no. 24) Largely lectures presented during a conference held at the

Mathematisches Forschungsinstitut Oberwolfach, Jan. 23-29, 1983. Includes bibliographies. 1. Noetherian rings—Congresses. 2. Universal enveloping algebras-

Congresses. 3. Lie algebras-Congresses. I. Small, Lance W., 1941 — II. Mathematisches Forschungsinstitut Oberwolfach. III. Series. QA251.4.N64 1987 512'.55 87-14997 ISBN 0-8218-1525-3 (alk. paper)

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Contents

Preface vii

The Goldie rank of a module J. T . STAFFORD l

Representation theory of semisimple Lie algebras THOMAS J. ENRIGHT 21

Primitive ideals in the enveloping algebra of a semisimple Lie algebra J . C. JANTZEN 29

Primitive ideals in enveloping algebras (general case) R. RENTSCHLER 37

Filtered Noetherian rings J A N - E R I K B J O R K 59

Noetherian group rings: An exercise in creating folklore and intuition DANIEL R. FARKAS 99

Preface

Noetherian rings abound in nature! Universal enveloping algebras of finite-dimensional Lie algebras and group algebras of polycyclic-by-finite groups are, for example, two wide classes of Noetherian rings. Yet, until Goldie's theorem was proved about thirty years ago, the "Noetherianness" of various types of noncommutative rings was not really effectively exploited.

Goldie's results provide the link between Noetherian rings and the much more studied case of Artinian rings. For instance, if R is a prime, right Noetherian ring, then R has a "ring of fractions" Q(R) which is of the form D n , n x n matrices over D a division ring. This last integer n is called the Goldie rank of R and plays an important role in some of the lectures in this collection. Later on the reader will meet some results involving Goldie ranks called "additivity principles" which arose, first, in studying rings satisfying a polynomial identity and then in enveloping algebras. For the moment, let's just state a very special case of such a result for semisimple Artinian rings: If Dn D Ami © • • • 0 Am ; , where D and the A's are division rings, then n = J2j=i zjmj where the z's are positive integers. All sorts of useful and, occasionally, tantalizing generalizations will be seen in the lectures of Stafford and Jantzen.

Five of the six expository lectures collected in this volume were presented at a conference on Noetherian rings at the Mathematisches Forschungsinstitut, Oberwolfach during the week of January 23-29, 1983. The other article by Thomas J. Enright is an introduction to the representation theory of finite-dimensional semisimple Lie algebras which he gave at a London Mathematical Society Durham conference in July, 1983. Enright's paper will serve as an excel­lent complement to the surveys of Jantzen and Rentschler. One of the purposes of the Oberwolfach meeting was to bring together specialists in Noetherian ring theory and workers in other areas of algebra who use Noetherian methods and results. The meeting was organized by Professors Walter Borho and Alex Rosen­berg and me.

If there is any common theme in these lectures, it is the study of the prime and primitive ideal spectra of various classes of rings: enveloping algebras, group algebras, polynomial identity rings, etc. A particularly useful tool is the Goldie rank of the ring factored by the relevant primes. Such investigations lead to the "additivity principles" discussed in Stafford's article. The Goldie rank then

vii

Vlll PREFACE

figures into the development of a convincing theory of localization in noncom-mutative rings which is described at the end of Stafford's paper.

The contributions of Jantzen and Rentschler treat the problem of classifying the primitive ideals of the universal enveloping algebras of finite-dimensional Lie algebras; Jantzen treats the semisimple case while Rentschler surveys the general situation. These papers are rather technical and the reader might also want to look at [1], for example.

Many of the methods developed for studying enveloping algebras can be suc­cessfully applied to the study of rings of differential operators on algebraic vari­eties and filtered algebras, more generally. J.-E. Bjork provides a careful devel­opment of certain aspects of filtered Noetherian rings culminating in an easy-to-understand proof of Gabber's theorem on the integrability of the characteristic variety. Bjork also provides background on rings of differential operators which have recently been a subject of great interest to ring theorists and analysts alike.

The concluding essay by D. R. Farkas focuses on the structure of group rings of polycyclic-by-finite groups (still the only known examples of Noetherian group algebras) discussing the Nullstellensatz, prime ideals a la Roseblade and those parts of the subject where Noetherian methods have proved valuable.

We turn now to some suggestions for additional reading on some of the more recent developments in Noetherian ring theory. A subject touched on in many of the lectures is affine algebras (i.e., algebras finitely generated over a field). The study of these algebras has grown immensely in recent years. A useful tool for investigating such algebras is Gelfand-Kirillov dimension. A clear, complete account of G-K dimension can be found in [4].

Localization theory continues to develop; there have been a number of "pay­offs," see, for example, [2] and [3]. Jategaonkar's monograph [5] is now available and provides an indispensable, though occasionally idiosyncratic, treatment of the state-of-art in localization techniques. Borho's lectures [1] also give interest­ing interpretations of certain localization techniques in the context of enveloping algebras.

Finally, what must surely become the standard reference for Noetherian rings, Robson and McConnell, is about to appear [6].

Lance W. Small

PREFACE IX

REFERENCES

1. W. Borho, A survey on enveloping algebras of semisimple Lie algebras, I, Can. Math. Soc. Conference Proceedings, Vol. 5, AMS, Providence, 1986.

2. K. A. Brown and R. B. Warfield, Jr., Krull and global dimensions of fully bounded Noetherian rings, Proc. Amer. Math. Soc. 92 (1984), 169-174.

3. K. R. Goodearl and L. W. Small, Krull versus global dimension in Noetherian PI rings, Proc. Amer. Math. Soc. 92 (1984), 175-178.

4. G. R. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics 116, Pitman, London, 1985.

5. A. V. Jategaonkar, Localization in Noetherian rings, London Math. Soc. Lecture Notes, Series 98, Cambridge, 1986.

6. J. C. McConnell and J. C. Robson, Rings with maximum condition, John Wiley and Sons, to appear.