minus one times an element is the
additive inverse in a ring∗
alozano†
2013-03-21 17:10:44
Lemma 1. Let R be a ring (with unity 1) and let a be an element of R. Then
(−1) · a = −a
where −1 is the additive inverse of 1 and −a is the additive inverse of a.
Proof. Note that for any a in R there exists a unique “−a” by the uniqueness
of additive inverse in a ring. We check that (−1) · a equals the additive inverse
of a.
a + (−1) · a
=
1 · a + (−1) · a,
by the definition of 1
=
(1 + (−1)) · a,
by the distributive law
=
0 · a,
=
0,
by the definition of − 1
as a result of the properties of zero
Hence (−1) · a is “an” additive inverse for a, and by uniqueness (−1) · a = −a,
the additive inverse of a. Analogously, we can prove that a · (−1) = −a as
well.
∗ hMinusOneTimesAnElementIsTheAdditiveInverseInARingi
created: h2013-03-21i by:
halozanoi version: h35674i Privacy setting: h1i hTheoremi h16-00i h13-00i h20-00i
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