Let $R$ be a commutative ring with $1$ and $M_n(R)$ be a group of $n \times n$ matrices over $R$.
$1$. Assume that $R_1 \cong R_2$ (as rings isomorphic). What is the relation between $M_n(R_1)$ and $M_n(R_2)$ as an $R_1-$module and an $R_2-$module, respectively?
$2$. Assume that $R \cong R_1 \times R_2$ (as rings isomorphic and the sum is external).
What can we say about $M_n(R), M_n(R_1)$ and $M_n(R_2)$ as modules?
I would like to view those modules via some type of module isomorphisms.