A note on monoform modules
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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. \begin{enumerate} \item[(1)] The prime $R$-module $M$ such that for any nonzero submodule $N$ of $M$, ${\rm{Ann}}_R(M/N)\neq{\rm{Ann}}_R(M)$. \item[(2)] Strongly prime $R$-module. \item[(3)] Faithful multiplication module over an integral domain. \end{enumerate}
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. \begin{enumerate} \item[(1)] The prime $R$-module $M$ such that for any nonzero submodule $N$ of $M$, ${\rm{Ann}}_R(M/N)\neq{\rm{Ann}}_R(M)$. \item[(2)] Strongly prime $R$-module. \item[(3)] Faithful multiplication module over an integral domain. \end{enumerate}
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