uniqueness of additive identity in a ring

Lemma 1.

Let $R$ be a ring. There exists a unique element $\mathrm{0}$ in $R$ such that for all $a$ in $R$:

$$0+a=a+0=a.$$

Proof.

By the definition of ring, there exists at least one identity in $R$, call it $0_1$. Suppose $0_2\in R$ is an element which also the of additive identity. Thus,

$$0_2+0_1=0_2$$

On the other hand, $0_1$ is an additive identity, therefore:

$$0_2+0_1=0_1+0_2=0_1$$

Hence $0_2=0_1$, i.e. there is a unique additive identity.∎