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Re: Finite Rings
Posted:
Feb 4, 2013 8:02 AM
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On Monday, February 4, 2013 10:26:16 AM UTC+2, William Elliot wrote:
> On Sun, 3 Feb 2013, Arturo Magidin wrote:
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> > On Sunday, February 3, 2013 10:46:14 PM UTC-6, William Elliot wrote:
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> > > On Sun, 3 Feb 2013, Arturo Magidin wrote:
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> > > > On Sunday, February 3, 2013 9:21:19 PM UTC-6, William Elliot wrote:
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> > > > > > If R is a finite commutative ring without multiplicative identity
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> > > > > > and if every element is a zero divisor, then does there exist
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> > > > > > a nonzero element which annihilates all elements of the ring?
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> > Which means that the trivial ring DOES NOT satisfy the hypothesis, and
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> > therefore is not to be considered, period. The fact that a ring without
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> > multiplicative identity must contain a nonzero element need not be a
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> > premise, because "does not have a multiplicative identity" IMPLIES,
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> > necessarily, the existence of a nonzero identity.
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> Clever but no better. According to John Beachy in "Abstract Algebra,"
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> the multiplicative identity is distinct from the additive identity.
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> So rather that toss bull about, what definitions are you using?
That Beachy example may not be that relevant here since many authors require
a ring NOT to be the zero ring, which is tantamount to requiring that the
neutral element and the multiplicative unit are different in the ring.
The fact here is that _IF_ we allow the zero ring, then it fulfills the
condition of having a multiplicative unit and, thus, cannot be taken as an
example.
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