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International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 531-534

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 Hindawi Publishing Corporation. 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.


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 xR is the sum of a potent element and a nilpotent element, then N and P are ideals and R=NP. We also prove that if R satisfies () and if each xR has a representation in the form x=a+u, where aP and uJ(R) ,then P is an ideal and R=J(R)P.