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

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Posts: 11,749
Registered: 12/4/04
Re: Finite Rings
Posted: Feb 3, 2013 11:25 PM
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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.

> So add the premise that R has a nonzero element.

Or, perhaps, not.

Arturo Magidin

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