Zero Matrix is Identity for Matrix Entrywise Addition over Ring

Theorem
Let $\struct {R, +, \circ}$ be a monoid whose identity is $e$.

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

Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the zero matrix of $\map {\MM_R} {m, n}$

Then $\mathbf 0_R$ is the identity element for matrix entrywise addition.

Proof
Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.

Then:

Mutatis mutandis, the same proof can be used to show that $\mathbf 0_R + \mathbf A = \mathbf A$.

Also see

 * Zero Matrix is Identity for Hadamard Product