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

Let $\mathrm{\Omega }$ be a group with identity $e$, $\mathrm{\Gamma }$ be a $\mathrm{\Omega }$-graded commutative ring and $\mathrm{\Im }$ a graded $\mathrm{\Gamma }$-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 $\mathrm{\Im }$ is said to be a $gr$-$C$-$2^A$-secondary submodule if whenever $r,s\in h(\mathrm{\Gamma })$, $L$ is a graded submodule of $\mathrm{\Im }$, and $rsS\subseteq L$, then either $rS\subseteq L$ or $sS\subseteq L$ or $rs\in Gr(An{n}_{\mathrm{\Gamma }}(S))$.