On generalized periodic-like rings. (Q925360)

Summary: Let \(R\) be a ring with center \(Z\), Jacobson radical \(J\), and set of nilpotent elements \(N\). Call \(R\) generalized periodic-like if for all \(x\in R\setminus(N\cup J\cup Z)\), there exist positive integers \(m,n\) of opposite parity for which \(x^m-x^n\in N\cap Z\). We identify some basic properties of such rings and prove some results on commutativity.