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

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 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Finite Rings
Posted: Feb 4, 2013 3:26 AM

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?