On graded $S$-comultiplication modules

TL;DR

This paper introduces graded S-comultiplication modules, establishing key properties and characterizations, including the relationship between submodules and graded S-prime ideals, advancing the understanding of graded module theory.

Contribution

It defines graded S-comultiplication modules and provides new characterizations linking submodules to graded S-prime ideals, expanding the theoretical framework.

Findings

N is a graded S-second submodule iff Ann_R(N) is a graded S-prime ideal and certain containment conditions hold.

Established equivalence conditions for submodules in graded S-comultiplication modules.

Provided foundational results for further research in graded module theory.

Abstract

In this paper, we introduce the concept of graded $S$-comultiplication modules. Several results concerning graded $S$-comultiplication modules are proved. We show that $N$ is a graded $S$-second submodule of a graded $S$-comultiplication $R$-module $M$ if and only if $Ann_R(N)$ is a graded $S$-prime ideal of $R$ and there exists $x\in S$ such that $xN\subseteq\bar{x}N$ for every $\bar{x}\in S$.