# Noncommutative Rings

I. N. Herstein
Volume: 15
Edition: 1
Pages: 215
https://www.jstor.org/stable/10.4169/j.ctt5hh9cw

1. Front Matter
(pp. i-viii)
2. PREFACE
(pp. ix-x)
(pp. xi-xii)
4. CHAPTER 1 THE JACOBSON RADICAL
(pp. 1-38)

This chapter has as its major goal the creation of the first steps needed to construct a general structure theory for associative rings. The aim of any structure theory is the description of some general objects in terms of some simpler ones—simpler in some perceptible sense, perhaps in terms of concreteness, perhaps in terms of tractability. Of essential importance, after one has decided upon these simpler objects, is to find a method of passing down to them and to discover how they weave together to yield the general system with which we began.

In carrying out such a program...

5. CHAPTER 2 SEMISIMPLE RINGS
(pp. 39-68)

The aim in defining the radical was to concentrate the bothersome behavior of a ring in a piece of it such that when this piece was removed the resulting ring was well enough behaved to permit some delicate dissection. The guidelines we choose for this dissection are the beautiful theorems of Wedderburn for the case of Artinian rings.

The success of this scheme is capable of a somewhat objective measure in the results that eventually come forth. We now undertake a more minute study of semi-simple rings. In the material developed we shall provide ourselves with an assortment of instruments...

6. CHAPTER 3 COMMUTATIVITY THEOREMS
(pp. 69-88)

In the preceding two chapters we laid out a general line of attack on ring-theoretic problems. This procedure is especially efficacious in proving that appropriately conditioned rings are commutative, or almost so. The reason for this lies in the trivial fact that a subring of a direct product of commutative rings is itself commutative. In the theorems to be considered we shall see clearly the role assumed by this general structure theory in the disposition of the problems at hand. Most particularly they illustrate effectively the point made earlier that in this kind of approach the difficulties usually present themselves...

7. CHAPTER 4 SIMPLE ALGEBRAS
(pp. 89-123)

Wedderburn’s pioneer work on the structure of simple algebras set the stage for deep investigations—often with an eye to application in algebraic number theory—in the theory of algebras. Much of the early research, following on the heels of that of Wedderburn, came in the work of Dickson. Then in the 1920’s and early 1930’s a very deep investigation of simple algebras was carried out culminating in a beautiful structure theory for division algebras over algebraic number fields. A large part of the results was developed in the hands of Albert, Artin, Brauer, Noether and many others.

As is...

8. CHAPTER 5 REPRESENTATIONS OF FINITE GROUPS
(pp. 124-149)

In Maschke’s theorem (Theorem 1.4.1) we showed that the group algebraF(G) of a finite groupGof ordero(G) over a fieldFof characteristic 0 orpwherepło(G) is semi-simple. By the theorems we have already proved about the nature of semi-simple Artinian rings the structure ofF(G) is fairly decisively pinned down. The information we garner this way aboutF(G) allows us to probe more deeply inGitself. It is this interplay betweenGandF(G) and its consequences that we propose to study in this chapter.We shall assume throughout—unless otherwise...

9. CHAPTER 6 POLYNOMIAL IDENTITIES
(pp. 150-168)

As we have indicated earlier on several occasions the general structure theory developed in the first two chapters reveals itself most effectively in the study of rings which are further restricted by some sort of polynomial condition. A good instance of this was seen in the study of the commutativity of rings.

We now pass to a class of rings—in some sense they satisfy a higher form of commutativity—which on analysis are capable of a sharp description. These rings are subjected to the presence of some nonzero polynomial relation in noncommuting variables, which vanishes identically on them. The...

10. CHAPTER 7 GOLDIE’S THEOREM
(pp. 169-186)

Goldie has recently proved several theorems which give penetrating information about the nature of rings subject to certain chain conditions. In the theory of rings with ascending chain conditions these theorems assume the role played by the Wedderburn theorems in the theory of rings with descending chain conditions. In fact these results are extensions of the Wedderburn theorems to a wider class of rings. Procesi gave a short and highly conceptual proof of the first of Goldie’s results; we then showed how one can easily obtain the second from the first.

The proof of these theorems that we shall present...

11. CHAPTER 8 THE GOLOD-SHAFAREVITCH THEOREM
(pp. 187-194)

This last, and very short, chapter has one central result and some of its noteworthy implications. This result is due to Golod and Shafarevitch, published recently in a remarkable paper. Their theorem, rather easy to prove, provides a general method and technique for considering a large assortment of problems. As an immediate consequence of the main theorem one can construct a nil but not locally nilpotent algebra thereby giving a negative answer to the Kurosh Problem; one can construct a torsion group which is not locally finite thereby settling in the negative the general Burnside Problem. In addition the theorem...

12. SUBJECT INDEX
(pp. 195-198)
13. NAME INDEX
(pp. 199-200)
14. Afterword
(pp. 201-202)
Lance W. Small

Noncommutative Ringsis a classic. It is fair to say that almost every practicing ring theorist has, at some time, studied portions of this book. Herstein’s style and grace make ring theory especially attractive. In this reprinting we have not changed text—only corrected typos. There are no additional chapters or remarks in the body of the text.

However, we do want to add a few remarks as a guide to the current literature and an indication of some of the enormous development in ring theory sinceNoncommutative Ringsappeared.

The theory of rings satisfying a polynomial identity had a...