Ring of Square Matrices over Field is Ring with Unity

Theorem
Let $F$ be a field.

Let $\map {\mathcal M_F} n$ be the $n \times n$ matrix space over $F$.

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

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

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

Proof
We have by definition that a field is a division ring which is also commutative, and hence a commutative ring with unity.

So, from Square Matrices forms Ring with Unity we have that $\struct {\map {\mathcal M_F} n, +, \times}$ is a ring with unity, but is not a commutative ring.

Hence the result.

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.