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
asked 13 hours ago
13571
235
add a comment |
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
asked 13 hours ago
13571
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
add a comment |
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
asked 13 hours ago
13571
235
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
asked 13 hours ago
13571
235
asked 13 hours ago
13571
235
asked 13 hours ago
13571
235
asked 13 hours ago
13571
235
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
add a comment |
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
add a comment |
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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