A weak periodicity condition for rings. (Q2575944)

An associative ring \(R\) is called a semi-weakly periodic ring if \(R\setminus(J(R)\cup Z)\subseteq P+N\), where \(Z\) is the center, \(N\) is the set of all nilpotent elements, \(J(R)\) is the Jacobson radical and \(P=\{x\in R\mid x=x^{n(x)},\;n(x)>1\}\). The authors investigate some interesting structure properties and commutativity conditions of such rings and prove the following main results for an arbitrary semi-weakly periodic ring \(R\neq J(R)\): (i) if \(N\) is commutative, then \(N\) is an ideal; (ii) if \(N\subseteq Z\), then \(R=Z\), (iii) if \(N\) is commutative and each element of \(R\setminus(J(R)\cup Z)\) is uniquely expressible as a sum of a potent element and a nilpotent, then \(R=Z\).