# Properties of Matrix Entrywise Addition over Ring

## Theorem

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {R, +, \circ}$.

Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$.

Let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation of matrix entrywise addition satisfies the following properties:

$+$ is closed on $\map {\MM_R} {m, n}$
$+$ is associative on $\map {\MM_R} {m, n}$
$+$ is commutative on $\map {\MM_R} {m, n}$.

## Proof

### Matrix Entrywise Addition over Ring is Closed

Let $\struct {R, +, \circ}$ be a ring.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.

For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation $+$ is closed on $\map {\MM_R} {m, n}$.

### Matrix Entrywise Addition over Ring is Associative

Let $\struct {R, +, \circ}$ be a ring.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.

For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation $+$ is associative on $\map {\MM_R} {m, n}$.

That is:

$\paren {\mathbf A + \mathbf B} + \mathbf C = \mathbf A + \paren {\mathbf B + \mathbf C}$

for all $\mathbf A$, $\mathbf B$ and $\mathbf C$ in $\map {\MM_R} {m, n}$.

### Matrix Entrywise Addition over Ring is Commutative

Let $\struct {R, +, \circ}$ be a ring.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.

For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation $+$ is commutative on $\map {\MM_R} {m, n}$.

That is:

$\mathbf A + \mathbf B = \mathbf B + \mathbf A$

for all $\mathbf A$ and $\mathbf B$ in $\map {\MM_R} {m, n}$.