Graded $r$-Submodules

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 $An{n}_M(a)=\{0\}$ (resp., $An{n}_R(x)=\{0\}$), 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.