Ring of Square Matrices over Commutative Ring with Unity

Theorem
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.

However, $\left({\mathcal M_R \left({n}\right), +, \times}\right)$ is not in general a commutative ring.

Proof
From Ring of Square Matrices over Ring with Unity we have that $\left({\mathcal M_R \left({n}\right), +, \times}\right)$ is a ring with unity.

However, Matrix Multiplication is Not Commutative.

Hence $\left({\mathcal M_R \left({n}\right), +, \times}\right)$ is not in general a commutative ring.

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.