Structure of weakly periodic rings with potent extended commutators

Abstract

A well-known theorem of Jacobson (1964, page 217) asserts that a ring R with the property that, for each x in R, there exists an integer n(x)>1 such that xn(x)=x is necessarily commutative. This theorem is generalized to the case of a weakly periodic ring R with a “sufficient” number of potent extended commutators. A ring R is called weakly periodic if every x in R can be written in the form x=a+b, where a is nilpotent and b is “potent” in the sense that bn(b)=b for some integer n(b)>1. It is shown that a weakly periodic ring R in which certain extended commutators are potent must have a nil commutator ideal and, moreover, the set N of nilpotents forms an ideal which, in fact, coincides with the Jacobson radical of R.