# 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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### Lang's elementary divisors theorem is wrong

The problem comes from Serge Lang's Algebraic Number Theory: Proposition 27 Elementary divisors theorem. Let $M$ be a non-zero finitely generated projective module over a Dedekind ring $A$. Then ...
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### Steinitz Isomorphism Theorem for Non-Dedekind domains

Fix a Dedekind domain $R$ and fractional ideals $I, J$. It's a classical result by Steinitz that $I\oplus J \cong R \oplus IJ$ as $R$-modules. Question Does this still hold for non-Dedekind 1-...
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### In Dedekind domain, fractional ideals are invertible. How can we precisely compute the inverse of a given ideal in Dedekind domains [closed]

One approach I am looking at is in Dedekind domains, for any given ideal I, there exists an ideal J such that IJ is principal. We can compute this J and find the inverse of the product which is a ...
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### Is an integral domain with every ideal a product of prime ideals necessarily a Dedekind domain?

Question: Is an integral domain in which every proper ideal is a product of prime ideals necessarily a Dedekind domain? Please provide a reference. The first two sentences of the Wikipedia page for ...
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### Chinese Remainder Theorem and ideals generated in localizations

In Milne's notes on algebraic number theory (https://www.jmilne.org/math/CourseNotes/ANT.pdf), on page 51, Corollary 3.14 and 3.15 both used the argument "use Chinese Remainder Theorem and look ...
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### Counterexample of Divisibility of Ideals with Product of Ideals

Given a commutative ring $R$ with unity, we define for $I,J\subseteq R$ ideals $I\ \vert\ J\iff I\supseteq J$ $IJ=\{\sum_i a_ib_i:a_i\in I, b_i\in J\}$ For every commutative unitary ring $R$ it ...
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### Unique unramified ideal implies that the ramification index is equal to the degree of field extension in a galois extension

Given a Galois extension $K \supseteq \mathbb{Q}$, prove that if there is only one unramified prime number $p$ over $K$ then there is only one prime ideal $\mathfrak{p} \subseteq O_K$ containing $p$ ...
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### Can polynomial rings be classified as a specific type of ring like algebraic integers are classified as Dedekind domains??

In algebraic number theory, we would like to study rings of algebraic integers but sometimes they are not PIDs and thus they don't possess good properties. Because of this, we have introduced the ...
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### Is the integral closure of an integral domain a dedekind domain?

I am reading through Milne's number theory notes, and he shows that the ring of integers of a field is a dedekind domain. He also shows that the integral closure of a dedekind domain A in a finite ...
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### Ad hoc proof of the fact that the localization of a ring of integers at a nonzero prime ideal is a PID

(Lot and lot of) Context. I am preparing lecture notes for a course on modules I will teach next year, and I was hoping for finding a "not-too-complicated" proof of the fact that an ideal of ...
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### Dedekind domain, non PID, localization over maximal ideals.

I was going through the notes, where these two examples on Dedekind domain I couldn't prove. I don't know much about Dedekind domains. Can anyone answer or at least provide some materials and hints so ...
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### Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains

Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$. What is the Tor-dimension of $A\otimes_k B$? ...
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### Prime of order is regular iff its decomposition in the normalization is trivial.

It's from a statement in Algebraic Number Theory by Neukirch, page 92. Example 5 "One can show..." Let ${o}$ be a one-dimensional noetherian integral domain and $\tilde{o}$ be its ...
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### $M$ $\mathscr{R}$-submodule complete lattice $\iff$ $aL \subset M \subset a^{-1}L$, $a \in \mathscr{R}$

My question may be nestled in the sense that there might be confusion about several notions, please let me know if it is the case. In "The Arithmetic of Hyperbolic 3-manifolds" by Maclachlan ...
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### Splitting field of $x^4+2$

While learning Galois theory, I tried to construct a splitting field for the polynomial $x^4+2$ over $\mathbb{Q}$, but I am terribly stuck. Since $x^4+2$ is irreducible by Eisenstein's criterion, I ...
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### Can the decomposition of a principal ideal in a Dedekind domain contain a non-principal ideal as a factor?

If $I= \prod_{P_i\in Spec(R)}P_i$ is the decomposition into prime ideals of a principal ideal $I$ in a Dedekind domain $R$, can one of the $P_i$ be a non-principal ideal? I guess it can’t, but I don’t ...
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### Defining invertible ideal as projective module of rank one contained in field of fraction

For a Dedekind domain $A$ such that $\operatorname{frac}(A)=K$, we define invertible ideals to be $I\hookrightarrow K$ and rank one projective modules over $A$. (I think this definition is motivated ...
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### Divisibility of Ideals [closed]

I'm an undergraduate student, who's in my final semester in university. I have a research project, but the advisor isn't the best. He said why won't we develop the notion of divisibility of ideals in ...
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### Prove Set is Dedekind Cut

Let $G$ be a Dedekind cut. Show that $H = \{x \in \mathbb{Q} :$There exists $a \in Q_{>0}$ such that $-x-a \not \in G \}$ Prove that $H$ is a Dedekind cut. I'm not entirely sure where to begin ...
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### general theory of ideal class groups

I'm looking for good references for ideal class groups. I have studied ideal class groups in two different context; in Algebraic number theory and in Quaternion Algebra (as they have connections with ...
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### Number of ideal classes of $\mathbb{Z}\left[\frac{1+\sqrt{65}}{2}\right]$ [duplicate]

Let $K=\mathbb{Q}(\sqrt{65})$, whose number ring is $\mathcal{O}=\mathbb{Z}\left[\frac{1+\sqrt{65}}{2}\right]$. I know that the number of ideal classes in $\mathbb{Z}\left[\frac{1+\sqrt{65}}{2}\right]$...
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