“School of Mathematics”
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Paper IPM / M / 9537  


Abstract:  
Let R be a commutative ring with identity and let M be a unital Rmodule. A submodule
N of M is called a dense submodule, if
M=∑_{φ}φ(N) where φ runs over all the
Rmorphisms from N into M. An Rmodule M is called a
πmodule if every nonzero submodule is dense in M. This
paper makes some observations concerning prime modules and
πmodules over a commutative ring. It is shown that an
Rmodule 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 cosemisimple modules over any
ring and for all finitely generated modules over a principal ideal
domain (PID), the two concepts π and prime are equivalent.
Rings R over which the two concepts π and prime are
equivalent for all Rmodules are characterized. Also, it is
shown that if M is a πmodule 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 πmodule if and only if either M is a simple module or R
is a Dedekind domain and M is a faithful Rmodule.
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