PlanetMath: zero times an element is zero in a ring

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zero times an element is zero in a ring

(Theorem)

Lemma 1 Let be a ring with zero element 0 (i.e. 0 is the additive identity of ). Then for any element $a\in R$ we have $0\cdot a = a\cdot 0 = 0$ .

Proof.

$\displaystyle 0\cdot a$	$\displaystyle =$	$\displaystyle (0+0)\cdot a ,$ by definition of zero
	$\displaystyle =$	$\displaystyle 0\cdot a + 0\cdot a,$ by the distributive law

Thus $0\cdot a=0\cdot a + 0\cdot a$ . Let

be the additive inverse of $0\cdot a \in R$ . Hence:

$\displaystyle b+0\cdot a=b+(0\cdot a + 0\cdot a)$
$\displaystyle (b+0\cdot a)=(b+0\cdot a) + 0\cdot a$
$\displaystyle 0=0 + 0\cdot a$
$\displaystyle 0=0\cdot a$

as claimed. The proof of $a\cdot 0=0$ is done analogously. $\qedsymbol$

"zero times an element is zero in a ring" is owned by alozano.

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$0\cdot a=0$

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Cross-references: inverse, identity, additive, zero element, ring
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This is version 5 of zero times an element is zero in a ring, born on 2004-03-09, modified 2006-03-09.
Object id is 5673, canonical name is 0cdotA0.
Accessed 5066 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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