  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.
