# Ring of Square Matrices over Commutative Ring with Unity

## Contents

## Theorem

Let $R$ be a commutative 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.

However, for $n \ge 2$, $\struct {\map {\mathcal M_R} n, +, \times}$ is not a commutative ring.

## Proof

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

However, Matrix Multiplication on Square Matrices over Ring with Unity is not Commutative.

Hence $\struct {\map {\mathcal M_R} n, +, \times}$ is not a commutative ring for $n \ge 2$.

For $n = 1$ we have that:

\(\, \displaystyle \forall \mathbf A, \mathbf B \in \map {\mathcal M_R} 1: \, \) | \(\displaystyle \mathbf A \mathbf B\) | \(=\) | \(\displaystyle a_{11} b_{11}\) | where $\mathbf A = \begin {pmatrix} a_11 \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} b_11 \end {pmatrix}$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle b_{11} a_{11}\) | as $R$ is a commutative ring | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \mathbf {B A}\) |

Thus, for $n = 1$, $\struct {\map {\mathcal M_R} n, +, \times}$ *is* a commutative ring.

$\blacksquare$

## 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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 29$: Theorem $29.2$