On 2-absorbing submodules and their Amalgamations

Let $R_1$ and $R_2$ be commutative rings with $1\neq 0,\;M$ and $N$ be unitary $R_1-$module and $R_2-$module, respectively. $f:R_1\rightarrow R_2$ be a ring homomorphism and $\varphi: M\rightarrow N$ be an $R-$module homomorphism. This article studied $2-$absorbing submodule that is a generalization of the concept of prime submodule. Firstly, some characterizations of $2-$absorbing submodule are presented. Then, we examine the notion of $2-$absorbing submodule in amalgamation module $M \bowtie^{\varphi} JN$. We detected when $F\bowtie^\varphi JN$ is a $2-$absorbing submodule of $M \bowtie^{\varphi} JN$ by using the isomorphism $\frac{M \bowtie^{\varphi} JN}{F \bowtie^{\varphi} JN} \cong \frac{M}{F}$ and the homomorphism $p_{\gamma}: M \bowtie^{\varphi} JN \rightarrow \varphi(M)+JN$, where $J$ be an ideal of $R_2$ and $F$ be a submodule of $M.$