International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 63171, 10 pages
doi:10.1155/2007/63171
Research Article
On Semiabelian π-Regular Rings
School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai 264005, China
Received 18 February 2007; Revised 29 March 2007; Accepted 25 April 2007
Academic Editor: Howard E. Bell
Copyright © 2007 Weixing Chen. 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
A ring R is defined to be semiabelian if every idempotent of R is either right semicentral or left semicentral. It is proved that the set N(R) of nilpotent elements in a π-regular ring R is an ideal of R if and only if R/J(R) is abelian, where J(R) is the Jacobson radical of R. It follows that a semiabelian ring R is π-regular if and only if N(R) is an ideal of R and R/N(R) is regular, which extends the fundamental result of Badawi (1997). Moreover, several related results and examples are
given.