On multiplication fs-modules and dimension symmetry

In this paper, we first study fs-modules, i.e., modules with finitely many small submodules. We show that every fs-module with finite hollow dimension is Noetherian. Also, we prove that an R-module M with finite Goldie dimension, is an fs-module if, and only if, M = M_۱ \oplus M_۲, where M_۱ is semisimple and M_۲ is an fs-module with Soc(M_۲) \ll M. Then, we investigate multiplication fs-modules over commutative rings and we prove that the lattices of R-submodules of M and S-submodules of M are coincide, where S=End_R(M). Consequently, M_R and _SM have the same Krull (Noetherian, Goldie and hollow) dimension. Further, we prove that for any self-generator multiplication module M, to be an fs-module as a right R-module and as a left S-module are equivalent.

Keywords:

Small submodules , fs-modules , multiplication modules , dimension symmetry