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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 (Ã¢ÂˆÂ—) 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 xÃ¢ÂˆÂˆR is the sum of a potent element and a nilpotent element, then N and P are ideals and R=NÃ¢ÂŠÂ•P. We also prove that if R satisfies (Ã¢ÂˆÂ—) and if each xÃ¢ÂˆÂˆR has a representation in the form x=a+u, where aÃ¢ÂˆÂˆP and uÃ¢ÂˆÂˆJ(R) ,then P is an ideal and R=J(R)Ã¢ÂŠÂ•P

Direct sums ofJ-rings and radical rings