PlanetMath: minus one times an element is the additive inverse in a ring

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minus one times an element is the additive inverse in a ring

(Theorem)

Lemma 1 Let be a ring (with unity ) and let be an element of . Then

$\displaystyle (-1)\cdot a = -a$

where is the additive inverse of and is the additive inverse of .

Proof. Note that for any

there exists a unique “

” by the uniqueness of additive inverse in a ring. We check that $(-1)\cdot a$ equals the additive inverse of

$\displaystyle a+(-1)\cdot a$	$\displaystyle =$	$\displaystyle 1\cdot a + (-1)\cdot a,$ by the definition of $\displaystyle 1$
	$\displaystyle =$	$\displaystyle (1+ (-1))\cdot a,$ by the distributive law
	$\displaystyle =$	$\displaystyle 0\cdot a,$ by the definition of $\displaystyle -1$
	$\displaystyle =$	$\displaystyle 0,$ as a result of the properties of zero

Hence $(-1)\cdot a$ is “an” additive inverse for

, and by uniqueness $(-1)\cdot a = -a$ , the additive inverse of

. Analogously, we can prove that $a\cdot (-1) = -a$ as well. $\qedsymbol$

"minus one times an element is the additive inverse in a ring" is owned by alozano.

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$(-1)\cdot a= -a$

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Attachments:: law of signs under multiplication in a ring (Derivation) by alozano

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Cross-references: uniqueness of additive inverse in a ring, inverse, additive, unity, ring
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This is version 6 of minus one times an element is the additive inverse in a ring, born on 2004-03-09, modified 2005-11-24.
Object id is 5674, canonical name is 1cdotAA.
Accessed 3446 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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