$R$-module homomorphism

Question

Let be $R$ a ring, $M$ a $R$ -module, $M_{1}$ and $M_{2}$ two submodules of $M$ and $\phi: M_{1} \oplus M_{2} \rightarrow M$ homomorphism defined by $\phi\left(x_{1}, x_{2}\right)=x_{1}+x_{2}$

then $\ker\phi$ is isomorphic to the $R$ -module $M_{1} \cap M_{2} $

This is always true or only when $M$ is a finite $R-$module ?

Answer 1

It is always true. Actually, you always have the short exact sequence \begin{alignat}{23} 0\longrightarrow M_1\cap M_2&\longrightarrow &M_1\oplus M_2&\longrightarrow M_1+ M_2\longrightarrow 0\\ x&\longmapsto &(x,-x)\\ &&(x,y)&\longmapsto x+y \end{alignat}