# Questions tagged [dedekind-domain]

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.

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### Does locally DVR implies Dedekind Domain when it is 1-dimensional, semi-local domain but Noetherian not given

Let R be a semi-local integral domain of dimension 1 such that $\forall P \in Spec{R}$ such that $P \ne 0$ we have, $R_P$ to be a Discrete Valuation Ring. Then prove that $R$ is a Dedekind Domain? ...
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### Fractional ideals decomposition

Let $R$ be a dedekind domain, and $K$ be it's field of fractions. Also, a fractional ideal is a finitely generated sub-R-module of K. I am trying to prove that a fractional ideal $\underline a$ can ...
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### Find the composition of finitely generated module over Dedekind domain

I'm taking a course on commutative algebra and we learn this theorem: Every finitely generated module M over Dedekind domain A is direct sum of projective module P and torsion module T. T is direct ...
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### Prime decomposition of pR in $\mathbb{A}\cap \mathbb{Q}[\alpha]$ for $\alpha={^3\sqrt{hk^2}}$ if p is a prime such that $p^2|m$

I'm going through Marcus number Field chapter 3 an I'm finding very hard to understand the part about the decomposition of pR (theorem 27) that tells us that if $p\not||R/Z[\alpha ]|$ then we can ...
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### Two equivalent definitions of Dedekind domains

In many algebra books, there exists the characterization of Dedekind domains. Some of them consists of at least 5 statements. However, I want to show the equivalance of the following statements: $R$ ...
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### Need help with an operation on finite length $A$-modules and Grothendieck Group

I'm reading Serre's local field, chapter 1 section 5, Norm and Inclusion Homomorphisms. In particular, he defines the operation $\chi_A$ from the category of finite length $A$-modules to the ...
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### 'Contain is to divide' doesn't imply Dedekind Domain

Let $A$ be a Containment-Division Ring $(\operatorname{CDR})$, i.e., an integral domain that satisfies that for all $I,J$ ideals of $A$ such that $I\subseteq J$, then $I=JK$ for some ideal $K$, that ...
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### Ideal quotient (or ideal saturation) and p-adic valuation.

It is from Serre's local field. Let $a$ and $b$ be two fractional ideals of Dedekind domain $A$, then $v_p((a:b))=v_p(a)-v_p(b)=v_p(ab^{-1})$. I can understand the last equality without difficulty. ...
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### Annihilator of a torsion module over a Dedekind domain

Let $A$ be a Dedekind domain and let $T$ be a finitely generated torsion $A$-module. Let $I:=\operatorname{Ann}(T)$. It is well known that $I$ has a unique decomposition as a product of prime ideals ...
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### Proving “to contain is to divide” for Dedekind domains

I'm currently reading Number Fields by Marcus and I'm trying to complete a proof left as an exercise. We have the statement as If A and B are ideals in a Dedekind domain R, then A|B iff A $\supset$ ...
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### Example of a Dedekind Domain which is not a PID

I am asked to show that $\mathbb R[X,Y]/(X^2+Y^2-1)$ is a DD but not a PID. Some quick observations I made are it is Noetherian, Normal (since $X^2-1$ is square free). How do I show the following ...
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### characterization of Dedekind domains with fractional Ideals [duplicate]

I know that Dedekind domains can be characterized as follows: $A$ is Dedekind iff every nonzero fractional ideal in A is invertible. (def of fractional ideal: $I \subset Frac(A)$ is finitely ...
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### Successive quotients of powers of a non-zero prime ideal in Dedekind domain

Let $R$ be a Dedekind domain, and $P$ a non-zero proper prime ideal of $R$. It is easy to show that we have proper descending chain of ideals $$R \supset P \supset P^2 \supset P^3\cdots$$ Also $R/P$ ...
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### powers of prime ideals in a Dedekind domain

Why in a Dedekind domain, $p^r \neq p^{r + 1}$ for any prime ideal $p$ and integer $r$?
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### Looking for an example of a non PIR commutative ring with every ideal two generated

I am looking for an example, with a direct proof, of a commutative ring with unity , which is not a Principal Ideal ring and every ideal is generated by at most $2$ elements. Any example or proof I ...
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### Let $K$ be an algebraic number field. Prove that $O_K$ contains infinitely many prime ideals. [duplicate]

Let $K$ be an algebraic number field. Prove that $O_K$ (collection of integral elements) contains infinitely many prime ideals. Now this is Dedekind domain. Now what should be the next step?
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### Cancellation property holds in ideals in Dedekind domains

Let $D$ be a Dedekind domain. Let $A,B,C$ be ideals of $D$ with $A\neq 0$ and $AB=AC$. Prove that $B=C$. I know that every fractional ideal is invertible here but what will happen with any ideal in ...
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### The norm function extends to a homomorphism

Let $R$ be a number ring that is Dedekind. Show that the norm function $N:I \to [R:I]$ on $R$-ideals extends to a homomorphism $N: \mathcal{I}(R) \to \mathbb{Q}^*$ where $\mathcal{I}(R)$ is the ...
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### Is there a finite quotient Dedekind domain with infinitely many primes of small norm?

A finite quotient Dedekind domain is a Dedekind domain $A$ such that $|A/I|$ is finite for every nonzero integral ideal $I$. Can it happen that, for some $d$, $|A/I|\leq d$ for infinitely many $I$? (...
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### Valuations on Dedekind domains and problem

I have multiples questions actually. Let's introduce the context. We consider $R$ a Dedekind ring, and the valuations $v_\mathfrak{p}$ associated with every $\mathfrak{p}$ prime ideal over $R$. In ...
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### If $I$ is a fractional ideal, $II^{-1}$ is an integral ideal ? (Dedekind ring)

My question is pretty simple, I don't figure out why, if $I$ is a fractional ideal of $A$ (where $A$ is a Dedekind ring), $II^{-1}$ (at the moment I do not have proved that $II^{-1} = A$) should be an ...