 
Summary: Result.Math. Online First
c 2008 Birkh¨auser Verlag Basel/Switzerland
DOI 10.1007/s0002500803055 Results in Mathematics
On Commutativity of Semiperiodic 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 semiperiodic if for each x R\(J Z), there exist
positive integers m, n of opposite parity such that xn
 xm
N. We inves
tigate commutativity of semiperiodic rings, and we provide noncommutative
examples.
Mathematics Subject Classification (2000). 16U80.
Keywords. Semiperiodic rings, commutativity, periodic rings, Chacron crite
rion.
1. Introduction
Let R be a ring with center Z = Z(R), Jacobson radical J = J(R), and set
N = N(R) of all nilpotent elements; and let Z and Z+
denote the ring of integers
and the set of positive integers. Define R to be periodic if for each x R, there
