Date: Feb 4, 2013 8:02 AM
Author: J. Antonio Perez M.
Subject: Re: Finite Rings
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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>

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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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>

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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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>

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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.