Matrix Entrywise Addition over Ring is Closed

Theorem
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 $\map {\MM_R} {m, n}$.

That is:
 * $\mathbf A + \mathbf B \in \map {\MM_R} {m, n}$

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