# Ring of Square Matrices over Ring with Unity

## Contents

## Theorem

Let $R$ be a ring with unity.

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

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

Then $\struct {\map {\mathcal M_R} n, +, \times}$ is a ring with unity.

## Proof

From Ring of Square Matrices over Ring is Ring we have that $\struct {\map {\mathcal M_R} n, +, \times}$ is a ring.

As $R$ has a unity, the unit matrix can be formed.

The unity of $\struct {\map {\mathcal M_R} n, +, \times}$ is this unit matrix.

$\blacksquare$

## Notation

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

- $\struct {\map {\mathcal M_R} n, +, \times}$

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

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.1$: Example $3$