Modules whose nonzero finitely generated submodules are dense

Abstract

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. First, we study multiplication $R$-modules $M$ where $R$ is a one dimensional Noetherian ring or $M$ is a finitely generated $R$-module. In fact, it is proved that if $M$ is a multiplication $R$-module over a one dimensional Noetherian ring $R$, then $Mcong I$ for some invertible ideal $I$ of $R$ or $M$ is cyclic. Also, a multiplication $R$-module $M$ is finitely generated if and only if $M$ contains a finitely generated submodule $N$ such that $Ann_R(N)= Ann_R(M)$. A submodule $N$ of $M$ is called dense in $M$, if $M=sum_varphivarphi(N)$ where $varphi$ runs over all the $R$-homomorphisms from $N$ into $M$ and $R$-module $M$ is called a weak $pi$-module if every non-zero finitely generated submodule is dense in $M$. It is shown that a faithful multiplication module over an integral domain $R$ is a weak $pi$-module if and only if it is a  Prufer prime module.