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

TL;DR

This paper introduces and studies a new class of graded submodules called $gr$-$C$-$2^{A}$-secondary submodules within graded modules over graded rings, exploring their properties and algebraic structure.

Contribution

It defines the concept of $gr$-$C$-$2^{A}$-secondary submodules and investigates their fundamental properties in the context of graded modules and rings.

Findings

Characterization of $gr$-$C$-$2^{A}$-secondary submodules

Conditions under which these submodules exhibit specific properties

Relationships with other classes of graded 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 $rs\,S\subseteq L$, then either $r\,S\subseteq L$ or $s\,S \subseteq L$ or $rs \in Gr(Ann_\Gamma(S))$.