Let $R$ be a commutative ring with identity $1\neq 0$ and $M$ be a non-zero unital $R$-module. In this paper, we generalize the concept of $r$-submodules of an $R$-module $M$ to the $I$-submodules of $M$ for ideals $I$ of $R$ with ${\rm ann}_{M}(I)=0$. ُConsider that $I$ is an ideal of $R$, a proper submodule $N$ of $M$ is an $I$-submodule if for every submodule $K$ of $M$, $IK\subseteq N$ with ${\rm ann}_{M}(I) = 0_{M}$, implies that $K\subseteq N$. Also, we say that a proper submodule $N$ of $M$ is a special $I$-submodule if for every submodule $L$ of $M$ the inclusion $IL\subseteq N$ with ${\rm ann}_{R}(L) = 0_{R}$, implies that $I\subseteq (N:M)$. We explore certain outcomes related to these categories of submodules.
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