Zero Matrix is Identity for Matrix Entrywise Addition over Ring/Proof 2

Theorem

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

Let $\map {\MM_R} {m, n}$ be a $m \times n$ 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

By definition, matrix entrywise addition is the Hadamard product with respect to ring addition.

We have from Ring Axiom $\text A3$: Identity for Addition that the identity element of ring addition is the ring zero $0_R$.

The result then follows directly from Zero Matrix is Identity for Hadamard Product.

$\blacksquare$