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ISRN Algebra
Volume 2012 (2012), Article ID 415207, 9 pages
http://dx.doi.org/10.5402/2012/415207
Research Article

On g-Semisymmetric Rings

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, AL-Azhar University, Nasr City, P.O. Box 11884, Cairo, Egypt

Received 22 February 2012; Accepted 15 March 2012

Academic Editors: H. Chen and F. U. Coelho

Copyright © 2012 Farahat S. Aly and Mohammed O. Al Mestady. 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

We introduce right (left) g-semisymmetric ring as a new concept to generalize the well-known concept: symmetric ring. Examples are given to show that these classes of rings are distinct. They coincide under some conditions. It is shown that 𝑅 is bounded right g-semisymmetric with boundary 1 from right if and only if 𝑅 is symmetric, whenever 𝑅 is regular. It is shown that a ring 𝑅 is strongly regular if and only if 𝑅 is regular and bounded right g-semisymmetric with boundary 1 from right. For a right 𝑝 . 𝑝 . -ring 𝑅 it is shown that 𝑅 is reduced if and only if 𝑅 is symmetric, if and only if 𝑅 is bounded right g-semisymmetric ring with boundary 1 from left, if and only if 𝑅 is IFP, if and only if 𝑅 is abelian. We prove that there is a special subring of the ring of 3 × 3 matrices over a ring without zero divisors which is bounded right g-semisymmetric with boundary 2 from left and boundary 2 from right. Also we show that flat left modules over bounded left g-semisymmetric ring with boundaries 1 from left and 1 from right are bounded left g-semisymmetric with boundaries 1 from left and 1 from right.