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multiplication 0 and 1
A bound arena with added than one aspect and no aught divisors is a analysis arena (Special case: a bound basic area is a field)
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2. Relevant equations
3. The attack at a solution
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Let ##r in R setminus {0}##, and ascertain ##f : R setminus {0} to R setminus {0}## by ##f(x) = rx##. Now accept ##f(x) = f(y)##. Then ##rx = ry## or ##r(x-y)=0##, which implies ##x=y## back ##r neq 0## and there can be no aught divisors. Back ##R setminus {0}## is finite, we can achieve ##f## is, in accession to actuality injective (as we aloof demonstrated), surjective. Similarly, we can appearance map authentic by appropriate multiplication is bijective.
Given ##r in R setminus {0}##, there exists an ##x in R setminus {0}## such that ##f(x) = r## or ##rx=r##, which implies ##r## has a appropriate identity. We can additionally deduce that ##r## has a larboard identity, and back larboard and appropriate identities charge coincide, ##x## is ##r##'s character simpliciter. Therefore, every nonzero aspect has an identity. Now we appearance that these identities are in actuality the aforementioned (this is area it gets a little bearded and uncertain).
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Let ##r## and ##s## be nonzero elements and let ##x## and ##y## be their identity, respectively. Then ##rs = rs## or ##rs = rys## or ##r = ry## or ##rx = ry## or ##x=y##, area we acclimated the abandoning law, which holds back there are no aught divisors, several times. There is apparently a added absolute way to achieve ##x=y##, but I can't see it at present.
Therefore, ##R## has a multiplicative identity, denote it as a ##1##. Thus accustomed ##1##, there exists an ##x in R setminus {0}## such that ##f(x) = 1## or ##rx = 1##. Using the actuality that larboard and appropriate inverses coincide, ##x## is ##r##'s multiplicative identity.
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36 Horizontal Multiplication Facts Questions -- 6 by 0-12 (A) | multiplication 0 and 1[/caption]
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Multiplying by Anchor Facts 0, 1, 2, 5 and 10 (Other Factor 1 to ... | multiplication 0 and 1[/caption]
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Multiplication facts 0 and 1 » WorkSheetsDirect.com | multiplication 0 and 1[/caption]