On graded $S$-comultiplication modules

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 $\bar{x}\in S$.