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.