Ring of Square Matrices over Real Numbers

Theorem
Let $\mathcal M_\R \left({n}\right)$ 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 $\left({\mathcal M_\R \left({n}\right), +, \times}\right)$ is a ring with unity, but is not a commutative ring.

Proof
We have that the Real Numbers form Totally Ordered Field.

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

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.