Summary: On Generalized Periodic-like Rings
Howard E. Bell
and Adil Yaqub
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.
Let R be a ring; and let N = N(R), Z = Z(R) and J = J(R) denote respectively the
set of nilpotent elements, the center, and the Jacobson radical. As usual, we call R periodic
if for each x R, there exist distinct positive integers m, n such that xm
. In  we
defined R to be generalized periodic (g計) if for each x R \ (N Z)
() there exist positive integers m, n of opposite parity such that xm
We now define R to be generalized periodic-like (g計衍) if () holds for each x R\(NJZ).
Clearly, the class of g計衍 rings contains all commutative rings, all nil rings, all Jacobson
radical rings, all g計 rings, and some (but not all) periodic rings. It is our purpose to exhibit
some general properties of g計衍 rings and to study commutativity of such rings.