The Nilpotence of Nil Subrings

Abstract

1. It is well kinown that, in any associative ring R with minimal condition oii (say) left ideals, every nil left ideal is nilpotent (for the simplest lknown proof of tllis, due to R. Brauer, see [4, Theorem 13, p. 64]) ; more recently, Jacobson has shown ([5], and cf. also [4, Theorem 30, p. 71]) that every nil subring of R, besides also certain types of (nil) subsets not admitting the additioin alnd mnultiplication of R, must be nilpotent. We recall also [4, Theorem 29, p. 71] that, at least in the presence of a two-sided identity element, the minimal condition on left ideals implies the maximal condition oni these ideals. Our main result in this note is that, in any associative ring with maximal condition on nilpotent subrings, every nil subring must in fact be nilpotent. Indeed, we establish the stronger resuilt stated as Theorem 1 below; this may be compared with a result of Levitzki [6], who showed that, if R has maximal conditioni on both left and right ideals, then every nil subring of R is nilpotent (and that every nil ideal is nilpotent under a weaker, though more complicated, hypothesis; cf. also [7]). In a concluding section, we establisl-h some analogous results for Lie rings. We note first a trivial lemma, which serves as a starting-point for our arguments in both the succeeding sections: