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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Which of the following are Dedekind domains? [closed]

Could you please advise me which of the following is a Dedekind domain, which not and why? $a) \; \mathbb Z[1/3]$ $b) \; \mathbb Z[\sqrt{-5}]$ $c) \; \mathbb Z[x]$ $d) \; \mathbb C[x,y]/(y^2 - x^3 +...
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Valuation at a prime ideal

I'm having troubles understanding the following definition Can somebody help me understanding that definition? What is for example the power of an ideal?
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Is it true that if an ideal $I$ of ring $R$ can be denoted as the product of ideals $J$ and $K$ then $I \subseteq J$ and $I \subseteq K$?

I just proof-read a proof of someone, and in the proof the assumption is used that if $I$ is an ideal of a ring $R$ such that $I = JK$ for some other ideals $J$ and $K$, then $I \subseteq J$ and $I \...
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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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Structure theorem for modules over Dedekind domains

I've come across the structure Theorem for fin. gen. Modules over a Dedekind domain several times now. It was formulated to us the following way: Let $R$ be a Dedekind domain. For every element $\...
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Factorization into prime ideals in dedekind domain $\mathbb{C}[t]_{(t)}[x]/(x^3+x^2+t)$

$R = \mathbb{C}[t]_{(t)}[x]/(x^3+x^2+t)$ is a dedekind domain. Therefore every proper ideal $I$ can be written as a product of prime ideals. I want to find the factorization of $I = (t+x^2-x)$ into ...
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Finding class number of quadratic number field using Minkowski bound

My understanding of this is as follows: In the general case, one has a quadratic number field $F$, which is always of the form $\mathbb{Q}(\sqrt{d})$ for some square-free integer $d$. Minkowski ...
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Show that $R[X]/(f)$ is Dedekind domain

Let $R$ be a dedekind domain, $K$ the field of fractions of $R$ and $f \in R[X]$ irreducible as polynomial in $K[X]$ s.t. $(f,f') = (1) = R[X]$. I want to check that $S = R[X]/(f)$ is also a dedekind ...
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Localization of Dedekind Domains

I'm struggling with the following question: suppose that $K$ is a Dedekind Domain with ring of integers $\mathcal{O}_K$, and that we have an element $x\in\mathfrak{p}$ such that $x\not\in\mathfrak{p}^...
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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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Bogus proof that every ideal in a Dedekind domain is principal

Let $A$ be a Dedekind domain and $I$ a nonzero ideal of $A$. For every $a \in I$, $(a)$ is contained in $I$, so $I$ divides $(a)$ and there exists some ideal $J_a$ such that $(a)=IJ_a$. We have $$I=\...
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Reference request: some properties of Dedekind rings

I'm looking for a source where I can look up for the proofs of certain fancy statements regarding Galois extensions of fraction fields of Dedekind rings. That is let $R$ be a Dedekind ring with ...
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Localization of ring of integers at discriminant

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\alpha \in \mathcal{O}_K$ that has degree $n$ over $\mathbb{Q}$, and let $d = D_{K/\mathbb{Q}}(1, \alpha, ..., \alpha^{n-1})$. ...
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Separability in the Definition of Ramification

Maybe this question will be clearer to me once I've read more about algebraic number theory but conversely maybe an answer to this question will help me at doing so. Let $\mathcal{O}$ be a Dedekind ...
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Is $\mathbb{Z}[\sqrt{-5}]$ Dedekind? [closed]

It is known that $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind ring. I want to prove it but I don't know. Please tell me if you know.
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A - Dedekind domain with ideal class group of rank h. Let I ⊆ A be an ideal such that I^m is principal and gcd(m,h)=1. Prove that I is principal. [closed]

As in title. Of course it is obvious when m=1, but I dont know how to solve it for greater m. Thanks in advance for any hints.
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Exact Sequence with Ideal Class Group

Let $R$ be a Dedekind Ring and $K = \operatorname{Quot}(R)$. Let $\mathcal{I}_K$ be the ideal group and $C \ell_K$ the ideal class group. In a lecture in algebraic number theory, our professor ...
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Easy examples of integrally closed non-factorial domains?

I'm looking for some easy to prove examples of Noetherian domains which are not UFD but are integrally closed in their fraction field. The common examples I know (like $\mathbb Q[x,y]/(x^2+y^2-1)$ ...
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$\operatorname{Spec}(A[\frac{b}{a}]) \longrightarrow \operatorname{Spec}(A)$ is an open immersion when $A$ is a Dedekind domain.

Let $A$ be a Dedekind domain, $a, b \in A$ with $a \neq 0$. I wonder if there is an elegant proof (using properties of Dedekind domain, not Zariski's main theorem of course) to show that $\...
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Projective ideal in $\mathbb{Z}[\sqrt{-5}]$

This is a question from my past Qual "Set $R=\mathbb{Z}[\sqrt{-5}]$ and $I=(2,1+\sqrt{-5})$. Is $I$ prime? Is $I$ projective as an $R$-module?" Clearly $R/I \cong \mathbb{Z}/2\mathbb{Z}$, hence $I$ ...
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Non-maximal orders contains non-invertible ideals

Let $K$ be an algebraic number field and $L$ the algebraic integers in $K$. Assuming an order $O$ is defined as a subring of $L$ that is of finite index as a subgroup of the additive group $L$, I will ...
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Extension of $\mathbb{Q}$ by the positive $2^n$ roots of $3$

Consider the following extension $L$ of $\mathbb{Q}$: $L = \mathbb{Q}(\alpha_1, \alpha_2, \ldots)$, where $\alpha_t$ is the positive number such that $\alpha_t ^{2^t} = 3$, $t \in\mathbb{N}$. Show ...
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Is ring of $S$-integers a Dedekind domain?

For $\mathcal{O}_{K}$, the integer ring of a global field, we denote $S$ to be any set of primes of a global field $K.$ Let $$\mathcal{O}_{K,S}:=\{x\in K\mid v_{\mathfrak{p}}\geq 0\text{ for }\...
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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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Suppose that the prime $p$ is totally ramified in $O_{K_1}$ and unramified in $O_{K_2}$. Prove that $K_1 \cap K_2=\Bbb Q$.

Let $K_1$ and $K_2$ be algebraic number fields. Suppose that the prime $p$ is totally ramified in $O_{K_1}$ and unramified in $O_{K_2}$. Prove that $K_1 \cap K_2=\Bbb Q$. For unramified $<p>=...
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Prove that $P_1,\cdots,P_r$ are distinct prime ideals of $O_K$ with $<p>=P_1^{e_1}\cdots P_r^{e_r}$ and $N(P_i)=p^{deg f_i},i=1,2,\cdots,r$

Let $K=\Bbb Q(\theta)$ be an algebraic number field with $\theta \in O_K$. Let $p$ be a rational prime. Let $$f(x)=irr_{\Bbb Q}(\theta)\in \Bbb Z[x]$$ Let $\bar{}$ denote the natural map $:\Bbb Z[x] \...
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Prove that $a^{N(P)}-1 \equiv 0(\text{mod P})$.

Let $K$ be an algebraic number field and $O_K$ be the ring of integers. Let $P$ be a prime ideal in $O_K$. Let $a \in O_K$ be such that $P \nmid \langle a\rangle$. Prove that $a^{N(P)}-1 \equiv 0(\...
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Prove that there exist only finitely many integral ideals of $O_K$ to which $m$ belongs

Let $K$ be an algebraic number field and $O_K$ be the ring of integers. Let $m \in \Bbb Z\setminus \{0\}$. Prove that there exist only finitely many integral ideals of $O_K$ to which $m$ belongs. ...
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Let $P$ be a prime ideal in the ring of integers then $P \cap \Bbb Z$ is prime in $\Bbb Z$

Let $P$ be a prime ideal in the ring of integers $O_K$ then $P \cap \Bbb Z=<p>$ for some prime $p \in \Bbb Z$. So what I have to prove is that $P \cap \Bbb Z$ is prime in $\Bbb Z$. Now $ab \in ...
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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 ...
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Quotient of a dedekind domain in a numberfield by a primeideal is a perfect field

I have the following question. Is the quotient of a dedekind domain A in a numberfield K by a prime I always a perfect field? I understand it for the ring of integers in a finite (separable) extension ...
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Extension of linearly independent set to a basis of a free module over a quotient of a Dedekind domain [closed]

Let $A$ be a Dedekind domain and a non-zero ideal $N$; we consider the commutative ring $R = A/N$. Let $S$ be a linearly independent subset of $R^m$. Can $S$ necessarily be extended to a basis of $R^...
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How to show that the ring is a Dedekind domain [duplicate]

How to show that $\mathbb{R}[x, y] / \langle x^2 + y^2 - 1 \rangle $ is a Dedekind domain?

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