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
Canonical name	UniquenessOfAdditiveIdentityInARing
Date of creation	2013-03-22 14:14:06
Last modified on	2013-03-22 14:14:06
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Theorem
Classification	msc 13-00
Classification	msc 16-00
Classification	msc 20-00