In a Dedekind domain the powers of a prime ideal are contained only in other powers
I'm currentley working on Dedekind domains and the following statement seems true to me but I don't know how to prove it.
If $Pneq 0$ is a prime ideal in a Dedekind domain, then if $ P^nsubseteq I$ with $I$ an ideal we have $I=P^m $ for some $mleq n $.
Is this true? If it is, how do I prove it?
maximal-and-prime-ideals dedekind-domain
asked Nov 22 at 4:33
Natalio
332111
add a comment |
I'm currentley working on Dedekind domains and the following statement seems true to me but I don't know how to prove it.
If $Pneq 0$ is a prime ideal in a Dedekind domain, then if $ P^nsubseteq I$ with $I$ an ideal we have $I=P^m $ for some $mleq n $.
Is this true? If it is, how do I prove it?
maximal-and-prime-ideals dedekind-domain
asked Nov 22 at 4:33
Natalio
332111
What have you tried?
– Dzoooks
Nov 22 at 4:47
1
Are you allowed to use that non-zero ideals in Dedekind domains admit a unique factorization into maximal ideals?
– asdq
Nov 22 at 4:48
I tried induction and using the fact that every prime ideal is maximal. And yes, I'm allowed to use the prime factorization. I was wondering if I can say something like if a product of primes is contained in a product of primes then each prime in the first product is contained in a prime in the second, or something like that.
– Natalio
Nov 22 at 12:10
add a comment |
I'm currentley working on Dedekind domains and the following statement seems true to me but I don't know how to prove it.
If $Pneq 0$ is a prime ideal in a Dedekind domain, then if $ P^nsubseteq I$ with $I$ an ideal we have $I=P^m $ for some $mleq n $.
Is this true? If it is, how do I prove it?
maximal-and-prime-ideals dedekind-domain
asked Nov 22 at 4:33
Natalio
332111
I'm currentley working on Dedekind domains and the following statement seems true to me but I don't know how to prove it.
If $Pneq 0$ is a prime ideal in a Dedekind domain, then if $ P^nsubseteq I$ with $I$ an ideal we have $I=P^m $ for some $mleq n $.
Is this true? If it is, how do I prove it?
maximal-and-prime-ideals dedekind-domain
maximal-and-prime-ideals dedekind-domain
asked Nov 22 at 4:33
Natalio
332111
asked Nov 22 at 4:33
Natalio
332111
asked Nov 22 at 4:33
Natalio
332111
asked Nov 22 at 4:33
Natalio
332111
asked Nov 22 at 4:33
Natalio
332111
332111
What have you tried?
– Dzoooks
Nov 22 at 4:47
1
Are you allowed to use that non-zero ideals in Dedekind domains admit a unique factorization into maximal ideals?
– asdq
Nov 22 at 4:48
I tried induction and using the fact that every prime ideal is maximal. And yes, I'm allowed to use the prime factorization. I was wondering if I can say something like if a product of primes is contained in a product of primes then each prime in the first product is contained in a prime in the second, or something like that.
– Natalio
Nov 22 at 12:10
add a comment |
What have you tried?
– Dzoooks
Nov 22 at 4:47
1
Are you allowed to use that non-zero ideals in Dedekind domains admit a unique factorization into maximal ideals?
– asdq
Nov 22 at 4:48
I tried induction and using the fact that every prime ideal is maximal. And yes, I'm allowed to use the prime factorization. I was wondering if I can say something like if a product of primes is contained in a product of primes then each prime in the first product is contained in a prime in the second, or something like that.
– Natalio
Nov 22 at 12:10
What have you tried?
– Dzoooks
Nov 22 at 4:47
What have you tried?
– Dzoooks
Nov 22 at 4:47
1
1
Are you allowed to use that non-zero ideals in Dedekind domains admit a unique factorization into maximal ideals?
– asdq
Nov 22 at 4:48
Are you allowed to use that non-zero ideals in Dedekind domains admit a unique factorization into maximal ideals?
– asdq
Nov 22 at 4:48
I tried induction and using the fact that every prime ideal is maximal. And yes, I'm allowed to use the prime factorization. I was wondering if I can say something like if a product of primes is contained in a product of primes then each prime in the first product is contained in a prime in the second, or something like that.
– Natalio
Nov 22 at 12:10
I tried induction and using the fact that every prime ideal is maximal. And yes, I'm allowed to use the prime factorization. I was wondering if I can say something like if a product of primes is contained in a product of primes then each prime in the first product is contained in a prime in the second, or something like that.
– Natalio
Nov 22 at 12:10
add a comment |
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What have you tried?
– Dzoooks
Nov 22 at 4:47
1
Are you allowed to use that non-zero ideals in Dedekind domains admit a unique factorization into maximal ideals?
– asdq
Nov 22 at 4:48
I tried induction and using the fact that every prime ideal is maximal. And yes, I'm allowed to use the prime factorization. I was wondering if I can say something like if a product of primes is contained in a product of primes then each prime in the first product is contained in a prime in the second, or something like that.
– Natalio
Nov 22 at 12:10