Ring of Square Matrices over Commutative Ring with Unity

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

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

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

Then $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity.

However, for $n \ge 2$, $\struct {\map {\MM_R} n, +, \times}$ is not a commutative ring.

Proof
From Ring of Square Matrices over Ring with Unity we have that $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity.

However, Matrix Multiplication is not Commutative.

Hence $\struct {\map {\MM_R} n, +, \times}$ is not a commutative ring for $n \ge 2$.

For $n = 1$ we have that:

Thus, for $n = 1$, $\struct {\map {\MM_R} n, +, \times}$ is a commutative ring.