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
= xn
. In [3] we
defined R to be generalized periodic (g­p) if for each x R \ (N Z)
() there exist positive integers m, n of opposite parity such that xm
- xn
N Z.
We now define R to be generalized periodic-like (g­p­l) if () holds for each x R\(NJZ).
Clearly, the class of g­p­l rings contains all commutative rings, all nil rings, all Jacobson
radical rings, all g­p rings, and some (but not all) periodic rings. It is our purpose to exhibit
some general properties of g­p­l rings and to study commutativity of such rings.

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics