Semi M-Projective and Semi N-Injective Modules

• Journal title : Kyungpook mathematical journal
• Volume 56, Issue 1,  2016, pp.83-94
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2016.56.1.83
Title & Authors
Semi M-Projective and Semi N-Injective Modules
Hakmi, Hamza;

Abstract
Let M and N be modules over a ring R. The purpose of this paper is to study modules M, N for which the bi-module [M, N] is regular or pi. It is proved that the bi-module [M, N] is regular if and only if a module N is semi M-projective and $\small{Im({\alpha}){\subseteq}^{\oplus}N}$ for all $\small{{\alpha}{\in}[M,N]}$, if and only if a module M is semi N-injective and $\small{Ker({\alpha}){\subseteq}^{\oplus}N}$ for all $\small{{\alpha}{\in}[M,N]}$. Also, it is proved that the bi-module [M, N] is pi if and only if a module N is direct M-projective and for any $\small{{\alpha}{\in}[M,N]}$ there exists $\small{{\beta}{\in}[M,N]}$ such that $\small{Im({\alpha}{\beta}){\subseteq}^{\oplus}N}$, if and only if a module M is direct N-injective and for any $\small{{\alpha}{\in}[M,N]}$ there exists $\small{{\beta}{\in}[M,N]}$ such that $\small{Ker({\beta}{\alpha}){\subseteq}^{\oplus}M}$. The relationship between the Jacobson radical and the (co)singular ideal of [M, N] is described.
Keywords
Regular ring;Semi-potent ring;Radical Jacobson;Semi M-injective (N-projective) modules;direct M-injective (N-projective) modules;Endomorphism ring;
Language
English
Cited by
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