$S$-$M$-cyclic submodules and some applications

Year 2025, Volume: 37 Issue: 37, 44 - 58, 14.01.2025

Samruam Baupradist

https://doi.org/10.24330/ieja.1480269

Abstract

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 $s\in S$ and $f \in Hom_R(M,N)$ such that $As \subseteq f(M) \subseteq A$. 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.

Keywords

$M$-cyclic submodule, $S$-$M$-cyclic submodule, $M$-principally injective module, $S$-$M$-principally injective module, pseudo-$M$-principally injective module, $S$-pseudo-$M$-principally injective module

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