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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 3, 2013 11:46 PM
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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?

> > Ask-an-Algebraist
> > No - the trivial ring.

> Incorrect. The trivial ring *does* have a multiplicative identity. I'll let you figure out what it is and why it *does* satisfy the condition that 1x=x=x1 for all x in the ring.
> In fact, I'll give you three guesses.
> The first two don't count, though.

Your trivial ring isn't as trivial as my trivial ring because
your trivial ring is fancied up with a multiplicative identity.

> > So add the premise that R has a nonzero element.
> Or, perhaps, not.

Definitely so for OP asked about rings without multiplicative identities.

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