Ring of Square Matrices over Ring is Ring

Theorem
Let $R$ be a ring.

Let $\mathcal M_R \left({n}\right)$ be the $n \times n$ matrix space over $R$.

Let $+$ be the operation of matrix entrywise addition.

Let $\times$ be (temporarily) used to represent the operation of conventional matrix multiplication.

Then $\left({\mathcal M_R \left({n}\right), +, \times}\right)$ is a ring.

Proof
From Matrix Addition over Group forms Group we have that $\left({\mathcal M_R \left({n}\right), +}\right)$ is an abelian group, because $\left({R, +}\right)$ is itself an abelian group.

Similarly, it is clear that $\left({\mathcal M_R \left({n}\right), \times}\right)$ is a semigroup, as Matrix Multiplication is Closed and Matrix Multiplication is Associative.

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

Note
When referring to the operation of matrix multiplication in this context, we must have some symbol to represent this, so $\times$ does as well as any.

However, we do not use $\mathbf A \times \mathbf B$ for $\mathbf A \mathbf B$ as it is used for something completely different.