zero-divisors of a ring constitute an ideal











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1
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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


















  • 2




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










  • @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
    yesterday












  • @AnyAD when the commutative algebra tag is used, we usually assume the ring in question is commutative.
    – rschwieb
    10 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


















  • 2




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










  • @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
    yesterday












  • @AnyAD when the commutative algebra tag is used, we usually assume the ring in question is commutative.
    – rschwieb
    10 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 yesterday









13571

235




235








  • 2




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










  • @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
    yesterday












  • @AnyAD when the commutative algebra tag is used, we usually assume the ring in question is commutative.
    – rschwieb
    10 hours ago














  • 2




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










  • @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
    yesterday












  • @AnyAD when the commutative algebra tag is used, we usually assume the ring in question is commutative.
    – rschwieb
    10 hours ago








2




2




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




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












@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
yesterday






@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
yesterday














@AnyAD when the commutative algebra tag is used, we usually assume the ring in question is commutative.
– rschwieb
10 hours ago




@AnyAD when the commutative algebra tag is used, we usually assume the ring in question is commutative.
– rschwieb
10 hours ago















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