uniqueness of additive identity in a ring

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. ∎

Title uniqueness of additive identity in a ring UniquenessOfAdditiveIdentityInARing 2013-03-22 14:14:06 2013-03-22 14:14:06 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 13-00 msc 16-00 msc 20-00