PlanetMath: uniqueness of additive identity in a ring

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uniqueness of additive identity in a ring

(Theorem)

Lemma 1 Let be a ring. There exists a unique element 0 in such that for all in :

$\displaystyle 0+a=a+0=a.$

Proof. By the definition of ring, there exists at least one identity in

, call it

. Suppose $0_2\in R$ is an element which also the of additive identity. Thus,

$\displaystyle 0_2+0_1=0_2$

On the other hand,

is an additive identity, therefore:

$\displaystyle 0_2+0_1=0_1+0_2=0_1$

Hence

, i.e. there is a unique additive identity. $\qedsymbol$

"uniqueness of additive identity in a ring" is owned by alozano.

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Cross-references: identity, ring
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This is version 3 of uniqueness of additive identity in a ring, born on 2004-03-09, modified 2004-11-22.
Object id is 5676, canonical name is UniquenessOfAdditiveIdentityInARing2.
Accessed 2822 times total.

Classification:

AMS MSC:	13-00 (Commutative rings and algebras :: General reference works )
	16-00 (Associative rings and algebras :: General reference works )
	20-00 (Group theory and generalizations :: General reference works )

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