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