On $Gr$-$C$-$2^A$-secondary submodules

Abstract

Let $\Omega$ be a group with identity $e$, $\Gamma$ be a $\Omega$-graded commutative ring and $\Im$ a graded $\Gamma$-module. In this article, we introduce the concept of $gr$-$C$-$2^A$-secondary submodules and investigate some properties of this new class of graded submodules. A non-zero graded submodule $S$ of $\Im$ is said to be a $gr$-$C$-$2^A$-secondary submodule if whenever $r,s\in h(\Gamma )$, $L$ is a graded submodule of $\Im$, and $rsS\subseteq L$, then either $rS\subseteq L$ or $sS\subseteq L$ or $rs\in Gr(An{n}_{\Gamma }(S))$.