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 a field.

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