S-M-cyclic submodules and some applications

In this paper, we introduce the notion of S-M-cyclic submodules, which is a generalization of the notion of M-cyclic submodules. Let $M, N$ be right R-modules and S be a multiplicatively closed subset of a ring R. A submodule A of N is said to be an S-M-cyclic submodule, if there exist s∈ S and f ∈ HomR(M,N) such that As ⊆ f(M) ⊆ A. Besides giving many properties of S-M-cyclic submodules, we generalize some results on M-cyclic submodules to S-M-cyclic submodules. Furthermore, we generalize some properties of principally injective modules and pseudo-principally injective modules to S-principally injective modules and S-pseudo-principally injective modules, respectively. We study the transfer of this notion to various contexts of these modules.