On Generalized Periodic-Like Rings

Abstract

Let R be a ring with center Z, Jacobson radical J, and set N of all nilpotent elements. Call R generalized periodic-like if for all x∈R∖(N∪J∪Z) there exist positive integers m, n of opposite parity for which xm−xn∈N∩Z. We identify some basic properties of such rings and prove some results on commutativity.