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 ∼= 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 annr(n) = annr(m). a submodule n of m is called dense in m, if m = σ φ φ(n) where φ runs over all the r-homomorphisms from n into m and r-module m is called a weak π-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 π-module if and only if it is a prüfer prime module.