PlanetMath: law of signs under multiplication in a ring

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law of signs under multiplication in a ring

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Lemma 1 Let be a ring with unity, which we denote by . For all $x,y\in R$ :

$\displaystyle (-x)\cdot (-y)=x\cdot y$

where denotes the additive inverse of in .

Proof. Here we use the fact $(-1)\cdot a = -a$ for all $a \in R$ . First, we see that:

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

since, clearly, the additive inverse of

itself.

Hence:

$\displaystyle (-x)\cdot (-y)=(-1)\cdot x \cdot (-1)\cdot y= (-1)\cdot (-1) \cdot x\cdot y= x \cdot y$

where we have used several times the associativity of $\cdot$ and the fact that $(-1)\cdot x = x \cdot (-1) = -x$ . $\qedsymbol$

"law of signs under multiplication in a ring" is owned by alozano.

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See Also: ring

Other names:

$(-x)\cdot (-y)= x\cdot y$

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Cross-references: associativity, inverse, additive, ring with unity
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This is version 7 of law of signs under multiplication in a ring, born on 2004-03-09, modified 2006-03-10.
Object id is 5675, canonical name is XcdotYXcdotY.
Accessed 3434 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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