Mathematics

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.

426 questions
30
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!
Truong
  • 4,342
26
votes
1 answer

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 there exists ideals $a_i, i= 1, \dots, r$ such that…
Functor
  • 1,757
19
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
  • 9,174
18
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 $h \neq k$,…
user14174
17
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 $M$ is…
D_S
  • 36,371
13
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…
user74489
  • 507
13
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…
Servaes
  • 68,935
12
votes
2 answers

A UFD is a PID if it has dimension $\le 1$ (nonzero prime ideals are maximal)

I am looking for 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. Here one dimensional means that every nonzero prime ideal is maximal. (Note: I have…
user233193
  • 123
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
  • 6,468
11
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
  • 924
  • 5
  • 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.
Tusif Ahmed
  • 536
  • 2
  • 17
9
votes
1 answer

If the norm of an ideal is prime then the ideal is prime

Let $K$ be a number field and $D = \mathcal{O}_K$ its ring of integers. Let $I$ be a non-zero integral ideal of $D$ and suppose that the norm $N(I)$ is a (rational) prime. I want to prove that $I$ is a prime ideal in $D$. Since $D$ is a Dedekind…
Edward Evans
  • 4,799
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 ?
user228169
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
  • 4,841
8
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
  • 5,427
1
2 3
…
28 29