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:


By the definition of ring, there exists at least one identityPlanetmathPlanetmath in R, call it 01. Suppose 02R is an element which also the of additive identity. Thus,


On the other hand, 01 is an additive identity, therefore:


Hence 02=01, 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