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.