# Dedekind domain finitely many prime ideals

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.

• Dedekind domains with finitely many primes are PIDs
• aic geometry Noetherian almost Dedekind domain MathOverflow

• 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

 Pulseras de cola de raton macrame knots By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. 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.
A Dedekind domain is a UFD if and only if its ideal class group is trivial.

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.