On commutativity in certain rings

Abstract

I.N. Herstein has shown that an associative ring in which the nilpotent elements are “well-behaved”, and such that every element satisfies a certain polynomial identity, is commutative. This result is generalized here. Specifically, it is shown that an alternative ring R which satisfies the following three properties is commutative:(i) for x ∈ R, there exists an integer n(x) and a polynomial px (t) with integer coefficients such that xn+1p(x) = xn;(ii) for a fixed positive integer m, a a nilpotent and b an arbitrary element of R, a - am commutes with b - bm;(iii) for the same m, a and b, (ab+b)m = (ba+b)m and (ab)m = ambm.Examples are given to show that all three properties are essential, and it is shown that for associative rings certain modified versions of these properties are individually enough to assure that the commutator ideal of the ring is nil.