26 - M. Behboodi, On prime modules and dense submodules, J. Commut. Algebra 4 (2012), 479-488 (ISI)
Let $R$ be a commutative ring with identity and let $M$ be a unital $R$-module. A submodule $N$ of $M$ is called a dense submodule, if $M=\sum_\varphi\varphi(N)$ where $\varphi$ runs over all the $R$-morphisms from $N$ into $M$. An $R$-module $M$ is called a
$\pi$-module if every non-zero submodule is dense in $M$. This paper makes some observations concerning prime modules and $\pi$-modules over a commutative ring. It is shown that an $R$-module $M$ is a prime module if and only if every nonzero cyclic submodule of $M$ is a dense submodule of $M$. Moreover, for modules with nonzero socles and co-semisimple modules over any ring and for all finitely generated modules over a principal ideal domain (PID), the two concepts $\pi$ and prime are equivalent. Rings $R$ over which the two concepts $\pi$ and prime are equivalent for all $R$-modules are characterized. Also, it is shown that if $M$ is a $\pi$-module over a domain $R$ with $dim(R)=1$, then either $M$ is a homogeneous semisimple module or a torsion free module. In particular, if $M$ is a multiplication module over a domain $R$ with $dim(R)=1$, then $M$ is a $\pi$-module if and only if either $M$ is a simple module or $R$ is a Dedekind domain and $M$ is a faithful $R$-module.