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Topic: Finite Rings
Replies: 28   Last Post: Feb 6, 2013 8:33 AM

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J. Antonio Perez M.

Posts: 2,736
Registered: 12/13/04
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:
>

> > On Sunday, February 3, 2013 10:46:14 PM UTC-6, William Elliot wrote:
>
> > > On Sun, 3 Feb 2013, Arturo Magidin wrote:
>
> > > > On Sunday, February 3, 2013 9:21:19 PM UTC-6, William Elliot wrote:
>
>
>

> > > > > > If R is a finite commutative ring without multiplicative identity
>
> > > > > > and if every element is a zero divisor, then does there exist
>
> > > > > > a nonzero element which annihilates all elements of the ring?
>
>
>

> > Which means that the trivial ring DOES NOT satisfy the hypothesis, and
>
> > therefore is not to be considered, period. The fact that a ring without
>
> > multiplicative identity must contain a nonzero element need not be a
>
> > premise, because "does not have a multiplicative identity" IMPLIES,
>
> > necessarily, the existence of a nonzero identity.
>
>
>
> Clever but no better. According to John Beachy in "Abstract Algebra,"
>
> the multiplicative identity is distinct from the additive identity.
>
>
>
> 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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