I want to know if
"zero-divisors of a ring constitute an ideal iff each pair of zero-divisors of the ring has a nonzero annihilator?"
the crucial point for zero-divisors of a ring to constitute an ideal is to check if the sum of each pair of zero-divisors is again a zero-divisor. so one direction is trivial:
if each pair of zero-divisors of the ring has a nonzero annihilator then
zero-divisors constitute an ideal.
what about the converse?
thanks.
Update
by ''each pair of zero-divisors has a nonzero annihilator'' i mean "for any pair of distinct zero-divisors like $a$ and $b$ we have a nonzero element $c\in R$ s.t. $ca=cb=0$.