# zero-divisors of a ring constitute an ideal

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. $$ca=cb=0$$.

• What do you mean with ''each pair of zero-divisors has a nonzero annihilator''? – Wuestenfux Nov 15 '18 at 8:38
• @13571 Is it a commutative ring? Otherwise, right or left annihilator? Also as commented above, your writing is not clear, 'pair' in what sense? – AnyAD Nov 15 '18 at 8:48
• @AnyAD when the commutative algebra tag is used, we usually assume the ring in question is commutative. – rschwieb Nov 16 '18 at 12:12
• by ''each pair of zero-divisors has a nonzero annihilator'' i mean "any distinct zero-divisors $a$ and $b$ has a nonzero common annihilator" – 13571 Nov 17 '18 at 8:21
• @rschwieb Thanks for that. – AnyAD Nov 18 '18 at 2:44