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.
Zhanmin Zhu1
, Zhangsheng Xia2
and Zhisong Tan2
1Department of Mathematics, Jiaxing University, Zhejiang, Jiaxing 314001, China
2Department of Mathematics, Hubei Institute for Nationalities, Hubei, Enshi 445000, China
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.