“School of Mathematics”
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| Paper IPM / M / 9537 |
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| Abstract: | |
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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=∑φφ(N) where φ runs over all the
R-morphisms from N into M. An R-module M is called a
π-module if every non-zero submodule is dense in M. This
paper makes some observations concerning prime modules and
π-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 π and prime are equivalent.
Rings R over which the two concepts π and prime are
equivalent for all R-modules are characterized. Also, it is
shown that if M is a π-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 π-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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