Structure of weakly periodic rings with potent extended commutators

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 x n ( 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 b n ( 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 .