Date: Feb 3, 2013 10:21 PM
Author: William Elliot
Subject: Finite Rings

> 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?


No - the trivial ring.
So add the premise that R has a nonzero element.