Date: Feb 3, 2013 11:25 PM
Author: magidin@math.berkeley.edu
Subject: Re: Finite Rings

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