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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.

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. Reference: Wikipedia.

It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition.

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In a Dedekind domain every ideal is either principal or generated by two elements.

Prove that in a Dedekind domain every ideal is either principal or generated by two elements. Help me some hints. Thanks a lot!
Truong
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Dedekind domain with a finite number of prime ideals is principal

I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. the ring is equal to $P_h + P_k$ for $h \neq k$,…
user14174
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If $A$ is a Dedekind domain and $I \subset A$ a non-zero ideal, then every ideal of $A/I$ is principal.

In this question I will use the following definition of a Dedekind domain: An integral domain $A$ is a Dedekind Domain if: 1) $A$ is a Noetherian Ring. 2) $A$ is integrally closed. 3) Every non-zero prime ideal of $A$ is maximal. I also know that…
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Finitely generated torsion module over a Dedekind domain

Let $M$ be a finitely generated torsion module over a Dedekind domain $R$. Show that there exist nonzero ideals $I_1 \supseteq \cdots \supseteq I_n$ of $R$ such that $M \cong \bigoplus\limits_{i=1}^n R/I_i$. I'm stuck on this problem. Since $M$…
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A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization $\mathcal{O}/\mathfrak{p}^n \cong…
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How does passing to ideals solve the problem of unique factorization?

$A:=\mathbb{Z}[\sqrt{-5}]$ is not a UFD, because for instance $$21 = 3 \cdot 7 = \left( 1+2\sqrt{-5}\right) \cdot \left(1-2\sqrt{-5}\right).$$ But since $A$ is a Dedekind domain, we should have unique factorization of ideals into prime ideals (this…
57Jimmy
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When is the tensor product of rings of integers again a ring of integers?

Given number fields $K$ and $L$, under what conditions does there exist a number field $M$ such that $$\mathcal{O}_K\otimes_{\Bbb{Z}}\mathcal{O}_L\cong\mathcal{O}_M.$$ It is necessary that $K$ and $L$ are linearly disjoint, but is this also…
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Ideal class group of $\mathbb{Q}(\sqrt{-65})$

I am trying to show that the ideal class group, $Cl(A)$ of $K:=\mathbb{Q}(\sqrt{-65})$, where $A$ is the ring of integers of $K$, is isomorphic to the product two cyclic groups of order 2 and 4, respectively. I am going to share the process I have…
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An example of prime ideal $P$ in an integral domain such that $\bigcap_{n=1}^{\infty}P^n$ is not prime

I am looking for an example of prime ideal $P$ in an integral domain such that the ideal $\bigcap_{n=1}^{\infty}P^n$ is not a prime ideal. This is a followup to this question where the ring was not assumed to be an integral domain.
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One-dimensional [Noetherian] UFD is a PID

I am looking for a reference which has a self-contained (elementary, that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does anyone know such a reference? (Note: I have looked at…
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Any commutative ring lying between a Dedekind domain and its fraction field is Dedekind?

Let $R$ be a commutative ring with unity, $D$ be a Dedekind domain, $K$ be its fraction field such that $D \subseteq R \subseteq K$. Then is it true that $R$ is Dedekind ?
user228169
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Detecting whether some rings are Dedekind domains

Consider the three rings $\mathbb{C}[x,y] / \langle x^4 + xy -1\rangle$, $\mathbb{Z}[x,y] / \langle x^4 + xy -1\rangle$ and $\mathbb{F}_2[x,y] /\langle x^4- y^3 \rangle$. I am supposed to detect whether these are Dedekind domains or not. However…
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Ideals in a non-dedekind domain that cannot be factored into product of primes

For a domain $R$ to be a Dedekind domain it need to satisfy 3 conditions: one-dimensional, Noetherian, integrally closed. I have got three domains satisfying all but one of those three: 1) $\mathbb{C}[x,y]$: not one-dimensional 2) Ring of all…
hxhxhx88
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An integral domain that is Noetherian, integrally closed, but not one-dimensional

A Dededind domain is defined as an integral domain which is integrally closed, one-dimensional, and Noetherian. Also I know an equivalent characterization that a domain is Dedekidn if and only if every nonzero ideals can factor into prime…
hxhxhx88
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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 $I$? (Equivalently, infinitely many primes.) I am…
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