International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 12, Pages 1853-1860
doi:10.1155/IJMMS.2005.1853

Generalizations of principally quasi-injective modules and quasiprincipally injective modules

Zhu Zhanmin,1 Xia Zhangsheng,2 and Tan Zhisong2

1Department of Mathematics, Jiaxing University, Zhejiang, Jiaxing 314001, China
2Department of Mathematics, Hubei Institute for Nationalities, Hubei, Enshi 445000, China

Received 22 November 2004; Revised 9 June 2005

Copyright © 2005 Zhu Zhanmin et al. 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 and M a right R-module with S=End(MR). The module M is called almost principally quasi-injective (or APQ-injective for short) if, for any m∈M, there exists an S-submodule Xm of M such that lMrR(m)=Sm⊕Xm. The module M is called almost quasiprincipally injective (or AQP-injective for short) if, for any s∈S, there exists a left ideal Xs of S such that lS(Ker(s))=Ss⊕Xs. In this paper, we give some characterizations and properties of the two classes of modules. Some results on principally quasi-injective modules and quasiprincipally injective modules are extended to these modules, respectively. Specially in the case RR, we obtain some results on AP-injective rings as corollaries.