International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 531-534
doi:10.1155/S0161171295000664
Direct sums of J-rings and radical rings
Department of Mathematics, Claina University of Mining and Technology, Jiangsu, Xuzhou 221008, China
Received 4 October 1993; Revised 20 May 1994
Copyright © 1995 Xiuzhan Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let R be a ring, J(R) the Jacobson radical of R and P the set of potent
elements of R. We prove that if R satisfies (∗) given x, y in R there exist integers
m=m(x,y)>1 and n=n(x,y)>1 such that xmy=xyn and if each x∈R is
the sum of a potent element and a nilpotent element, then N and P are ideals and R=N⊕P. We also prove that if R satisfies (∗) and if each x∈R has a representation
in the form x=a+u, where a∈P and u∈J(R) ,then P is an ideal and R=J(R)⊕P.