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 identityPlanetmathPlanetmath in R, call it 01. Suppose 02R is an element which also the of additive identity. Thus,

02+01=02

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

02+01=01+02=01

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