Zero Matrix is Identity for Hadamard Product

Theorem
Let $\left({S, \circ}\right)$ be a monoid whose identity is $e$.

Let $\mathcal M_S \left({m, n}\right)$ be an $m \times n$ matrix space over $S$.

Let $\left[{e}\right]_{m n}$ be the zero matrix of $\mathcal M_S \left({m, n}\right)$

Then $\left[{e}\right]_{m n}$ is the identity element for matrix entrywise addition.

Proof
Let $\left[{a}\right]_{m n} \in \mathcal M_S \left({m, n}\right)$.

Then $\forall \left({i, j}\right) \in \left[{1 \,.\,.\, m}\right] \times \left[{1 \,.\,.\, n}\right]: a_{i j} \circ e = a_{i j} = e \circ a_{i j}$.