Ring of Square Matrices over Commutative Ring with Unity

Let $$R$$ be a commutative ring with unity.

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 with unity.

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.


 * The unity is the Identity Matrix.


 * 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.