δ-M-Small and δ-Harada Modules

Abstract

Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N≅ K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δ _M -lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained.

Key Words:

• Harada module and ring
• Injective module
• Lifting module
• Noetherian QF-module
• Small module

2000 Mathematics Subject Classification:

• 16L30
• 16E50

ACKNOWLEDGMENT

Authors would like to express their gratefulness to Professor Yiqiang Zhou (Memorial University of Newfoundland) for valuable suggestions and the referee for careful reading and several comments that improved the article.

Notes

Communicated by R. Wisbaner.