zero-divisors of a ring constitute an ideal











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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.






commutative-algebra ideals







share|cite|improve this question


















  • 1




    What do you mean with ''each pair of zero-divisors has a nonzero annihilator''?
    – Wuestenfux
    13 hours ago










  • @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
    13 hours ago

















up vote
1
down vote

favorite












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.






commutative-algebra ideals







share|cite|improve this question


















  • 1




    What do you mean with ''each pair of zero-divisors has a nonzero annihilator''?
    – Wuestenfux
    13 hours ago










  • @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
    13 hours ago















up vote
1
down vote

favorite









up vote
1
down vote

favorite











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.






commutative-algebra ideals







share|cite|improve this question













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.






commutative-algebra ideals




commutative-algebra ideals






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 13 hours ago









13571

235




235








  • 1




    What do you mean with ''each pair of zero-divisors has a nonzero annihilator''?
    – Wuestenfux
    13 hours ago










  • @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
    13 hours ago
















  • 1




    What do you mean with ''each pair of zero-divisors has a nonzero annihilator''?
    – Wuestenfux
    13 hours ago










  • @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
    13 hours ago










1




1




What do you mean with ''each pair of zero-divisors has a nonzero annihilator''?
– Wuestenfux
13 hours ago




What do you mean with ''each pair of zero-divisors has a nonzero annihilator''?
– Wuestenfux
13 hours ago












@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
13 hours ago






@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
13 hours ago

















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