Modules whose nonzero finitely generated submodules are dense

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 M\cong 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_\varphi\varphi(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.

Keywords:

Dense submodules , Multiplication modules , Prime modules , Weak pi-modules