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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. $ca=cb=0$.

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

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