> > 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?
Assume without debate nor dispute, that R has a non-zero element p.
If there's no nonzero annihilator, then for all nonzero x, there's some a_x with nonzero a a_x.
Let p0 = p and uning induction, for all j in N, define p_(j+1) = pj a_pj. p0, p1, p2, ... is sequence of nonzero elements.
Since R is finite, there's some distinct j,k with pj = pk. Thusly we've some nonzero a,b with ab = a. Now what?