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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.
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