On 2-absorbing primary submodules of modules over commutative rings

All rings are commutative with $1\neq0$, and all modules are unital. The purpose of this paper is to investigate the concept of $2$-absorbing primary submodules generalizing $2$-absorbing primary ideals of rings. Let $M$ be an $R$-module. A proper submodule $N$ of an $R$-module $M$ is called a $2$-absorbing primary submodule of $M$ if whenever $a,b\in R$ and $m\in M$ and $abm\in N$, then $am\in M$-$rad(N)$ or $bm\in M$-$rad(N)$ or $ab\in(N:_RM)$. It is shown that a proper submodule $N$ of $M$ is a $2$-absorbing primary submodule if and only if whenever $I_1I_2K\subseteq N$ for some ideals $I_1,I_2$ of $R$ and some submodule $K$ of $M$, then $I_1I_2\subseteq(N:_RM)$ or $I_1K\subseteq M$-$rad(N)$ or $I_2K\subseteq M$-$rad(N)$. We prove that for a submodule $N$ of an $R$-module $M$ if $M$-$rad(N)$ is a prime submodule of $M$, then $N$ is a $2$-absorbing primary submodule of $M$. If $N$ is a $2$-absorbing primary submodule of a finitely generated multiplication $R$-module $M$, then $(N:_RM)$ is a $2$-absorbing primary ideal of $R$ and $M$-$rad(N)$ is a $2$-absorbing submodule of $M$.