Ring of Square Matrices over Real Numbers

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

Let $\map {\mathcal M_\R} n$ be the $n \times n$ matrix space over the set of real numbers $\R$.

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

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

Then $\struct {\map {\mathcal M_\R} n, +, \times}$ is a ring with unity, but is not a commutative ring.

Proof
Recall that Real Numbers form Field.

The result follows directly from Ring of Square Matrices over Field is Ring with Unity.