r -Submodules and sr-Submodules

In this article, we introduce new classes of submodules called $r$-submodule and special $r$-submodule, which are two different generalizations of $r$-ideals. Let $M $be an $R$-module, where $R $is a commutative ring$. $We call a proper submodule $N\ $of $M$ an $r$-submodule (resp., special $r$-submodule) if the condition $am\in N$ with $ann_{M}(a)=0_{M} $(resp., $ann_{R}(m)=0$) implies that $m\in N$ (resp., $a\in(N:_{R} M)$) for each $a\in R $and $m\in M. $ We also give various results and examples concerning $r$-submodules and special $r$-submodules.