I. N. Herstein

Topics In Ring Theory

In this chapter we shall make a study of rings satisfying certain ascending chain conditions. In the non-commutative case-and this is really the only case with which we shall be concerned- the decisive and incisive results are three theorems due to Goldie. The main part of the chapter will be taken up with a presentation of these.

Lie ideals and derivations of prime rings

In this paper, we shall consider the relationship between the derivations and Lie ideals of a prime ring. Some of the results we obtain have been obtained earlier, even for rings more general than prime rings, in the case of inner derivations. We shall also look at the action of derivations on Lie ideals; the results we obtain extend some that had been proved earlier only for the action of derivations on the ring itself. Let R be a ring and d # 0 a derivation of R. If U is a Lie ideal of R, we shall be concerned about the size of d(C/). How does one measure this size? One way is to look at the centralizer of d(U) in R; the bigger d(U), the smaller this centralizer should be. This explains our interest in the centralizer of d(U). The result we obtain generalizes the principal theorem of [ 11. We may also measure the size of d(U) by looking at how large d(U), the subring generated by d(U), turns out to be. We view d(U) as large if it contains a non-zero ideal of R. For our special setting, we will obtain a result which generalizes one in [2]. Finally, a well-known and often used result states that if d is a derivation of R, which is semi-prime and 2-torsion-free, such that dZ = 0 then d = 0 (see the proof of Lemma 1.1.9 in [3]). If R is prime, of characteristic not 2, and d*(I) = 0 for a non-zero ideal, 1, of R, it also follows that d = 0. What can one say if d’(U) = 0 for some non-central Lie ideal of R? For inner derivations this was studied and answered in [4]. For prime rings and for any derivation d # 0 we answer the question of when d*(U) = 0 completely in our Theorem 1. We shall be working in the context of prime rings of characteristic not 2 in all that we do here. However, many of the results have some suitable analog for semi-prime, 2-torsion-free rings. In the presence of 2-torsion most of our results are not valid as stated, but something non-trivial can be said in

A proof of a conjecture of Vandiver

The Wedderburn theorem that every finite division ring is commutative has been extended by several authors [1].1 Vandiver, in his paper The p-adic representation of rings [2 ] conjectured the following generalization of Wedderburns theorem: every finite, non-commutative ring contains an element which is a divisor of zero and is not in the centrum. In this paper we give a short and simple proof of this conjecture. We also exhibit one generalization of it which was pointed out to us by the referee.

THE JACOBSON RADICAL

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 there are many paths one can follow, many classes of candidates for these simpler objects, and one must choose among these for that theory which is most fruitful in producing decisive results. In the case of rings there seems to be no doubt that the fundamental structure theory laid out by Jacobson is the appropriate one. The best proof of this remark is the host of striking theorems which have resulted from the use of these methods. Modules. Essential to everything that we shall discuss—in fact essential in every phase of algebra—is the notion of a module over a ring R or, in short, an R- module. To be absolutely precise we should say a right R -module for we shall allow the elements of R to act on the module from the right. However we shall merely say R-module, understanding by that term a right R- module.

On a conjecture on simple groups

The purpose of this paper is to rephrase a conjecture about simple groups into the language of linear algebra. Let G be a group of finite order o(G). Then by rF we shall mean the group ring of G over a field of characteristic p (for instance the integers modulo p). We shall denote the radical of rF by N,. If p = 0 or p o(G), then it is known that Np=(O); and if p|o(G), Np (O). We now consider the following two assertions: (A) If G is a simple group of odd order, o(G) is a prime. (B) If G is a group of odd order o(G), then for some prime p, p[ o(G), we can find a gCG, g1, such that g-1CNp. The theorem which we propose to prove is: