Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(2): 505-514

Online first article March 13, 2019      Printed March 1, 2019

https://doi.org/10.4134/BKMS.b180382

Copyright © The Korean Mathematical Society.

A note on monoform modules

Alireza Hajikarimi, Ali Reza Naghipour

Islamic Azad University; Monash University

Abstract

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. A submodule $N$ of $M$ is called a dense submodule if ${\rm{Hom}}_R(M/N,E_R(M))=0$, where $E_R(M)$ is the injective hull of $M$. The $R$-module $M$ is said to be monoform if any nonzero submodule of $M$ is a dense submodule. In this paper, among the other results, it is shown that any kind of the following module is monoform.
(1) The prime $R$-module $M$ such that for any nonzero submodule $N$ of $M$, ${\rm{Ann}}_R(M/N)\neq{\rm{Ann}}_R(M)$.
(2) Strongly prime $R$-module.
(3) Faithful multiplication module over an integral domain.

Keywords: dense submodule, prime module, monoform module, injective hull, multiplication module

MSC numbers: 13C05, 13C11, 13E05