Maximal ideals are always coprime. The last condition means precisely that every nonzero prime ideal is maximal, so maximality of nonzero primes is tautological. Viewed 4k times. Home Questions Tags Users Unanswered. MathOverflow works best with JavaScript enabled. Active 2 years, 10 months ago. In order to apply CRT we should assume that the prime ideals are coprime, i. Question feed. Noetherian almost Dedekind domain Ask Question. Email Required, but never shown.

Video: Dedekind domain finitely many prime ideals Unique Factorization of Ideals in Dedekind Domains

Here's one proof. Let R be a Dedekind ring and assume that the prime ideals are p1,pn. Then p21,p2,pn are coprime.

Pick an element π∈p1∖p21 and by. Dedekind Domains, Finitely Many Prime Ideals.

One Prime Ideal. Let R be a dedekind domain with one prime ideal P. All ideals are powers of P, and are linearly.

## Dedekind domains with finitely many primes are PIDs

ideal of B is, in particular, an A-submodule of B, hence is finitely generated over A and . A Dedekind domain with only finitely many prime ideals is a PID. Proof.

Bill Dubuque Bill Dubuque k 31 31 gold badges silver badges bronze badges. Mathematics Stack Exchange works best with JavaScript enabled. An almost Dedekind domain with noetherian maximal spectrum is a Dedekind domain. Viewed times. In the sense that the sum of two of them gives the entire ring.

## aic geometry Noetherian almost Dedekind domain MathOverflow

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Active 2 years, 6 months ago. Why did my reputation suddenly increase by points? Related 3. Andro Zimone Andro Zimone 11 2 2 bronze badges. I note that all of your answers are taking as granted that the ideals are maximal. |

show that every ideal is contained in only finitely many prime ideals.

the localizations of A at its nonzero prime ideals are all discrete . fractional ideal I we have vp(I)=0 for all but finitely many prime ideals p.

Suppose that R is a Dedekind domain such as the ring of algebraic integers in a number field. Although there are infinitely many prime ideals in.

Email Required, but never shown. Jason Juett Jason Juett 2 2 silver badges 5 5 bronze badges. Sign up using Facebook.

Why did my reputation suddenly increase by points? It is well known that a Dedekind domain is noetherian and so it's maximal spectrum is noetherian space as a subspace of Zariski topology. From the definition I read it is not immediate to me why those ideals are maximal.

In the sense that the sum of two of them gives the entire ring.

Asked 7 years, 11 months ago.

Active 2 years, 10 months ago. Noetherian almost Dedekind domain Ask Question.

Related 3.