A weak periodicity condition for rings

An element x of the ring R is called periodic if there exist distinct positive integers m, n such that xm = xn; and x is potent if there exists n > 1 for which xn = x. We denote the set of potent elements by P or P(R), the set of nilpotent elements by N or N(R), the center by Z or Z(R), and the Jacobson radical by J or J(R). The ring R is called periodic if each of its elements is periodic, and R is called weakly periodic if R = P +N . It is easy to show that every periodic ring is weakly periodic, but whether the converse holds is apparently not known. It has long been known that periodic rings have nice commutativity behavior; in particular, Herstein [10] showed that if R is periodic and N ⊆ Z, then R is commutative—a result which extends easily to weakly periodic rings. Various generalized periodic and weakly periodic rings have been introduced in recent years, and their commutativity behavior has been explored [6, 7, 13, 14]. Define R to be semi-weakly periodic if R\(J ∪ Z) ⊆ P +N . Clearly the class of semiweakly periodic rings is quite large; it contains all weakly periodic rings, all commutative rings, and all Jacobson radical rings. Our purpose is to point out some general properties of semi-weakly periodic rings and to investigate commutativity of such rings.