# Ring of Square Matrices over Field is Ring with Unity

## Contents

## 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 Commutativeā€ˇ, we have that $\struct {\map {\mathcal M_F} n, +, \times}$ is not a commutative ring.

Hence the result.

$\blacksquare$

## Notation

When referring to the operation of **matrix multiplication** in the context of the ring of square matrices:

- $\struct {\map {\MM_R} n, +, \times}$

we *must* have some symbol to represent it, and $\times$ does as well as any.

However, we do *not* use $\mathbf A \times \mathbf B$ for matrix multiplication $\mathbf A \mathbf B$, as it is understood to mean the vector cross product, which is something completely different.

## Sources

- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.2$: Some examples of rings: Ring Example $7$