Ring of Square Matrices over Field is Ring with Unity

Theorem
Let $F$ be a field.

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

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

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.

Hence $F$ is a commutative ring with unity.

So, from Ring of Square Matrices over Commutative Ring with Unity we have that $\struct {\map {\mathcal M_F} n, +, \times}$ is a ring with unity.

From Matrix Multiplication is not Commutative, we have that $\struct {\map {\mathcal M_F} n, +, \times}$ is not a commutative ring.

Hence the result.