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.

373 questions
27
votes
2 answers

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!
15
votes
3 answers

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 $h \neq k$,…
14
votes
2 answers

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…
Rankeya
  • 8,396
  • 1
  • 21
  • 58
14
votes
1 answer

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 $M$…
D_S
  • 30,734
  • 6
  • 38
  • 107
11
votes
2 answers

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…
11
votes
3 answers

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
  • 5,906
  • 8
  • 31
11
votes
1 answer

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…
10
votes
1 answer

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…
GSF
  • 896
  • 4
  • 15
10
votes
1 answer

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.
10
votes
2 answers

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…
9
votes
2 answers

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 ?
9
votes
2 answers

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…
Paul Slevin
  • 4,505
  • 15
  • 40
8
votes
2 answers

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…
7
votes
2 answers

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
  • 4,979
  • 22
  • 46
7
votes
2 answers

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
  • 4,979
  • 22
  • 46
1
2 3
24 25