Linked Questions
15 questions linked to/from Equivalent characterizations of discrete valuation rings
0
votes
2
answers
180
views
Questions of Proposition 9.2 from Atiyah's Introduction to Commutative Algebra
Proposition 9.2. Let $A$ be a Noetherian local domain of dimension one, $\mathfrak{m}$ its maximal ideal, $k = A / \mathfrak{m}$ its residue field.
Then the following are equivalent:
i) $A$ is a ...
- 1,128
3
votes
2
answers
1k
views
How to show that a valuation ring has a unique maximal ideal?
A subring $R$ of a fi eld $K$ is said to be a valuation ring of $K$ if for each
$x$ $\in$ $K^{*}$ we have either $x$ $\in$ R or $x^{-1}$ $\in$ $R$.
How can I show that the valuation ring has a unique ...
- 11.1k
3
votes
1
answer
455
views
Intuition behind discrete valuation rings
I'm trying to understand what DVRs are. I have seen two formulations, one in terms group homomorphisms and a discrete valuation function satisfying an axiom of the sort:
$$ v(x+y) \geq \text{min} \...
- 42k
5
votes
1
answer
219
views
Noetherian domain with unique principal prime ideal that is not a DVR
The question is whether such a thing exists. Namely, a discrete valuation ring (DVR), in whatever way you define it, is quite obviously a domain, Noetherian, and has a unique prime element up to ...
- 24.8k
1
vote
0
answers
67
views
Explanation on discrete valuation rings
Let $A$ be a noetherian domain. We want to prove that if $A$ is normal then, for every prime ideal $p$ associated to a principal ideal of $A$, the localization $A_p$ is a discrete valuation ring. It ...
- 76.9k
1
vote
0
answers
320
views
What makes a Principal Ideal Domain a Euclidean Domain?
The definitions of Principal Ideal Domain (PID) and Euclidean Domain (ED) are both from integral domain:
A non-trivial ring $R$ with no zero divisors is said to be entire; a
commutative entire ring ...
- 4,861
0
votes
0
answers
55
views
DVR is PID proof [duplicate]
I already know that if $R$ is a Noetherian local ring with Krull dimension $1$, then $R$ is DVR if and only if its maximal ideal $\mathfrak{m}$ is principal ideal if and only if every nonzero ideal is ...
- 4,672
36
votes
4
answers
6k
views
Intuitive explanation of Nakayama's Lemma
Nakayama's lemma states that given a finitely generated $A$-module $M$, and $J(A)$ the Jacobson radical of $A$, with $I\subseteq J(A)$ some ideal, then if $IM=M$, we have $M=0$.
I've read the proof, ...
- 1,547
3
votes
0
answers
141
views
Every Principal Ideal Domain is a Unique Factorization Domain
I am interested in verifying the existence aspect of the theorem asserting that every Principal Ideal Domain is a Unique Factorization Domain. In the first paragraph, I (think that I) have provided an ...
- 654
2
votes
1
answer
2k
views
Proof of every PID is Noetherian
I saw the proof of this proposition in here, but I have a question about this.
Definition of Noetherian ring is that ring is commutative, and every ideal of R is finitely generated, right? Principal ...
- 150k
4
votes
1
answer
2k
views
Krull Intersection Theorem
In this proof I know since R is noetherian it can be written as descending sequence of ideals which stabilizes after finite steps. Also I know since R is noetherian implies every ideal is finitely ...
- 51.6k
3
votes
1
answer
1k
views
Why is a P.I.D. Dedekind domain?
Definition
A Noetherian integrally closed domain of Krull dimension $1$ is said to be a Dedekind domain.
Since fields are of Krull dimension $0$, fields are not Dedekind domain. However, it is ...
CommunityBot
- 1
3
votes
1
answer
1k
views
The proof of Krull's Principal Ideal Theorem
Theorem: Let $R$ be Noetherian and $P$ be a minimal prime ideal over $(a)$ for some nonunit $a$ of $R$. Then $\operatorname{ht}(P)\leq 1$.
My lecture notes prove this as follows.
WLOG $R$ is local ...
- 51.6k
1
vote
1
answer
498
views
Characterization of Discrete Valuation Rings
Let $R$ be a Noetherian local domain with unique maximal ideal $M$. Then I want to show that if every $M$-primary ideal is a power of $M$, then $R$ is a Discrete Valuation Ring.
I know I'll be done ...
- 51.6k
5
votes
1
answer
2k
views
Deduce that a Noetherian valuation ring is either a field or a Discrete Valuation Ring.
I'm trying to solve this question from a book and I have already proved 1.
Let $R$ be a local domain which is not a field. Suppose that the maximal ideal $M$ of $R$ is principal and satisfies $\cap_{n=...
CommunityBot
- 1