# Ring of Square Matrices over Commutative Ring with Unity

## Theorem

Let $R$ be a commutative ring with unity.

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

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

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

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

## Proof

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

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

For $n = 1$ we have that:

 $\ds \forall \mathbf A, \mathbf B \in \map {\MM_R} 1: \,$ $\ds \mathbf A \mathbf B$ $=$ $\ds a_{11} b_{11}$ where $\mathbf A = \begin {pmatrix} a_11 \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} b_11 \end {pmatrix}$ $\ds$ $=$ $\ds b_{11} a_{11}$ as $R$ is a commutative ring $\ds$ $=$ $\ds \mathbf {B A}$

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

$\blacksquare$

## Notation

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

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

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.