Graded $r$-Submodules

TL;DR

This paper introduces and studies the concepts of graded r-submodules and graded special r-submodules in G-graded modules over commutative G-graded rings, generalizing graded r-ideals and exploring their properties.

Contribution

It defines new classes of graded submodules, investigates their properties, and provides illustrative examples, expanding the theory of graded modules.

Findings

Characterization of graded r-submodules and graded special r-submodules.

Identification of properties and behaviors of these new classes.

Examples illustrating the concepts and their distinctions.

Abstract

Let $G$ be a group with identity $e$ and $R$ a commutative $G$-graded ring with a nonzero unity $1$. In this article, we introduce the concepts of graded $r$-submodules and graded special $r$-submodules, which are generalizations for the notion of graded r-ideals. For a nonzero $G$-graded $R$-module $M$, a proper graded $R$-submodule $K$ of $M$ is said to be graded $r$-submodule (resp., graded special $r$-submodule) if whenever $a\in h(R)$ and $x\in h(M)$ such that $ax\in K$ with $Ann_{M}(a)=\{0\}$ (resp., $Ann_{R}(x)=\{0\}$), then $x\in K$ (resp., $a\in (K:_{R}M)$). We study various properties of graded $r$-submodules and graded special $r$-submodules, and we give several illustration examples of these two new classes of graded modules.