On Generalizations of Graded $2$-absorbing and Graded $2$-absorbing primary submodules

TL;DR

This paper introduces and studies generalized concepts of graded 2-absorbing and 2-absorbing primary submodules in graded modules over commutative rings, extending existing theories with new properties and characterizations.

Contribution

The paper defines graded -absorbing and -absorbing primary submodules via a -function, and explores their properties and relationships to existing submodule concepts.

Findings

Introduced graded -absorbing submodules with new properties.

Characterized graded -absorbing primary submodules.

Established relationships with graded radicals and existing submodules.

Abstract

Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. In this article, we introduce the concepts of graded $\phi$-$2$-absorbing and graded $\phi$-$2$-absorbing primary submodules as generalizations of the concepts of graded $2$-absorbing and graded $2$-absorbing primary submodules. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $\phi: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $\phi$-$2$-absorbing $R$-submodule of $M$ if whenever $x, y$ are homogeneous elements of $R$ and $m$ is a homogeneous element of $M$ with $xym\in K-\phi(K)$, then $xm\in K$ or $ym\in K$ or $xy\in (K :_{R} M)$, and $K$ is said to be a graded $\phi$-$2$-absorbing primary $R$-submodule of $M$ if whenever $x, y$ are homogeneous elements of $R$ and $m$ is a homogeneous element of $M$ with $xym\in K-\phi(K)$, then $xm$ or $ym$ is in the graded radical of $K$ or $xy\in (K:_{R}M)$. We investigate several properties of these new types of graded submodules.