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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.

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  1. If $R_1 \cong R_2$ then $M_n(R_1) \cong M_n(R_2)$ as abelian groups, as $R_1$-modules, as $R_2$-modules, as $R_1$-algebras, and as $R_2$-algebras.

  2. If $R \cong R_1 \times R_2$ then $M_n(R) \cong M_n(R_1) \times M_n(R_2)$ as abelian groups and as $R$-algebras.

Write an element of $M_n(R)$ as $M = (m_{i,j})$, $m_{i,j} \in R$ for $1 \leq i,j \leq n$. Write each $m_{i,j} = m^1_{i,j} + m^2_{i,j}$, $m^k_{i,j} \in R_k$ for $k=1,2$. And write $M^1 = (m^1_{i,j}) \in M_n(R_1)$, $M^2 = (m^2_{i,j}) \in M_n(R_2)$. The map $M \mapsto (M^1,M^2) \in M_n(R_1) \times M_n(R_2)$ is an $R$-algebra isomorphism.

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