Strongly Maximal Submodules with A Study of Their Influence on Types of Modules

Fatima M Mohialdeen,Buthyna N Shihab

doi:10.30526/35.1.2802

Abstract

Let S be a commutative ring with identity, and A is an S-module. This paper introduced an important concept, namely strongly maximal submodule. Some properties and many results were proved as well as the behavior of that concept with its localization was studied and shown.

Highlights

Along with this paper, S is a commutative ring with identity, and A is an S-module
From the fact that every maximal submodule is a semimaximal by [3, Remarks and Examples (2.1.2), p32], and the fact no.(11), we obtain that every SM-submodule of an Smodule A is a semimaximal while the converse is not true in general and the following shows that: Let 6Z be a submodule of a Z-module Z
P is a prime submodule of A, by [3, proposition (1.1.51)] we obtain that A/P is a prime S-module, and by corollary (3.6), A/P is a simple S-module and A/P has no SM-submodule

Summary

Introduction

S is a commutative ring with identity, and A is an S-module. A proper submodule N of an S-module A is named maximal if there exists a submodule D of A such that N⊊D⊆A, D=A[1][11]. Proposition (2.3) An S-module A is simple if and only if A ≅ S/E for some maximal ideal E of S [4] [1]. B is named strongly maximal submodule (for short SM-submodule) if and only if, for every non-zero ideal E of S implies A/E2B is a regular module.

Objectives

Results

Conclusion