uniqueness of additive identity in a ring∗
alozano†
2013-03-21 17:10:55
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
01 . Suppose 02 ∈ R 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.
∗ hUniquenessOfAdditiveIdentityInARingi created: h2013-03-21i by: halozanoi version:
h35676i Privacy setting: h1i hTheoremi h13-00i h16-00i h20-00i
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