Direct sums of \(J\)-rings and radical rings (Q1895281)

The paper deals with the rings \(R\) satisfying the following condition (*): For each \(x,y\in R\) there exist integers \(m=m(x,y)>1\) and \(n=n(x,y)>1\) such that \(x^my=xy^n\). Let \(J(R)\) be the Jacobson radical of \(R\), \(N\) the set of nilpotent elements of \(R\), and \(P=\{x\in R\mid x^n=x\) for a certain integer \(n=n(x)>1\}\) the set of potent elements of \(R\). The main results of the paper are as follows: (i) If \(R\) satisfies (*) then \(N\) and \(P\) are ideals and \(R=P\oplus N\); if, moreover, any \(x\in R\) is a sum \(a+u\), \(a\in P\), \(u\in J(R)\) then \(R=J(R)\oplus P\); (ii) A semisimple ring satisfying (*) is isomorphic to a subdirect sum of fields.