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.