Zero Matrix is Identity for Matrix Entrywise Addition over Ring/Proof 2
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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.
$\blacksquare$