Direct sums of J-rings and radical rings

Abstract

Let R be a ring, J(R) the Jacobson radical of R and P the set of potent elements of R. We prove that if R satisfies (∗) given x, y in R there exist integers m=m(x,y)>1 and n=n(x,y)>1 such that xmy=xyn and if each x∈R is the sum of a potent element and a nilpotent element, then N and P are ideals and R=N⊕P. We also prove that if R satisfies (∗) and if each x∈R has a representation in the form x=a+u, where a∈P and u∈J(R) ,then P is an ideal and R=J(R)⊕P.