Ring of Square Matrices over Ring is Ring

Theorem
Let $R$ be a ring.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\struct {\map {\mathcal M_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$.

Then $\struct {\map {\mathcal M_R} n, +, \times}$ is a ring.

Proof
From Matrix Entrywise Addition over Group forms Group we have that $\struct {\map {\mathcal M_R} n, +}$ is an abelian group, because $\struct {R, +}$ is itself an abelian group.

Similarly, it is clear that $\struct {\map {\mathcal M_R} n, \times}$ is a semigroup, as Matrix Multiplication is Closed and Matrix Multiplication is Associative.

Finally, we note that Matrix Multiplication Distributes over Matrix Addition.