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The Scientific World Journal
Volume 2013 (2013), Article ID 750808, 3 pages
http://dx.doi.org/10.1155/2013/750808
Research Article

The Unique Maximal GF-Regular Submodule of a Module

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 20 July 2013; Accepted 15 August 2013

Academic Editors: J. He, I. G. Ivanov, J. Rada, and L. Székelyhidi

Copyright © 2013 Areej M. Abduldaim and Sheng 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

An -module is called -regular if, for each and , there exist and a positive integer such that . We proved that each unitary -module contains a unique maximal -regular submodule, which we denoted by . Furthermore, the radical properties of are investigated; we proved that if is an -module and is a submodule of , then . Moreover, if is projective, then is a -pure submodule of and .