# Zero Matrix is Identity for Matrix Entrywise Addition over Ring

## 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 1

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

Then:

 $\ds \mathbf A + \mathbf 0_R$ $=$ $\ds \sqbrk a_{m n} + \sqbrk {0_R}_{m n}$ Definition of $\mathbf A$ and $\mathbf 0_R$ $\ds$ $=$ $\ds \sqbrk {a + 0_R}_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \sqbrk a_{m n}$ Ring Axiom $\text A3$: Identity for Addition is $0_R$ $\ds \leadsto \ \$ $\ds \mathbf A + \mathbf 0_R$ $=$ $\ds \mathbf A$ Definition of Zero Matrix over Ring

Similarly:

 $\ds \mathbf 0_R + \mathbf A$ $=$ $\ds \sqbrk {0_R}_{m n} + \sqbrk a_{m n}$ Definition of $\mathbf A$ and $\mathbf 0_R$ $\ds$ $=$ $\ds \sqbrk {0_R + a}_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \sqbrk a_{m n}$ Ring Axiom $\text A3$: Identity for Addition is $0_R$ $\ds \leadsto \ \$ $\ds \mathbf 0_R + \mathbf A$ $=$ $\ds \mathbf A$ Definition of Zero Matrix over Ring

$\blacksquare$

## Proof 2

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