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 a 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.