Lemma 1   Let $R$ be a ring. There exists a unique element $0$ 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.