William Elliot <email@example.com> wrote: > >[In forum "Ask an Algebraist", user "Anu" asks:] > >> 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.
Even with that correction, the answer is still "no".
Consider the commutative ring R consisting of the following seven distinct elements:
0, x, y, z, x+y, y+z, z+x
Besides the usual laws required for R to be a commutative ring (without identity), we also require the following relations:
r^2 = r for all r in R
r+r = 0 for all r in R
xy = yz = zx = 0
Note that the above relations imply
(x+y)z = (y+z)x = (z+x)y = 0
so every element of R is a zerodivisor.
However, since all elements of R are idempotent, it follows that no nonzero element of R annihilates all elements of R.