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Questions tagged [krull-dimension]

For questions about or related to the Krull dimension, which counts the length of the longest chain of prime ideals of a ring under inclusion.

The Krull dimension of a commutative ring $R$ is defined to be the supremum of the lengths of chains of prime ideals in $R$. Given a chain of prime ideals

$$p_0 \subsetneq p_1 \subsetneq \dots \subsetneq p_n$$

we define the length of this chain to be $n$ (that is, $n$ is the number of strict inclusions). The Krull dimension is the supremum of the quantity $n$ over all such chains.

A field has Krull dimension $0$, and any principal ideal domain that is not a field has Krull dimension $1$. It is not necessary that a ring has finite Krull dimension, even if the ring is Noetherian.

If $M$ is an $R$-module, we define the Krull dimension of $M$ to be

$$\dim_R M = \dim(R/\operatorname{Ann}_R(M))$$

where $\operatorname {Ann}_R(M)$ is the annihilator in $R$ of $M$.

Reference: Krull dimension.

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Classification of commutative rings $R$ satisfying $\dim(R[T])=\dim(R)+1$

Let $R$ be a commutative ring. Then $\dim(R[T]) \geq \dim(R)+1$. Is there a classification of those commutative rings with the property $\dim(R[T])=\dim(R)+1$? Every Noetherian commutative ring has this property, but also every $0$-dimensional…
Martin Brandenburg
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Ideal in polynomial ring which contains no non-zero prime ideal

Let $J$ be a non-zero ideal in $\mathbb C[X,Y]$ such that $J$ contains no non-zero prime ideal. Then is it true that $J$ has height $1$ ? Possible approach: Since $\mathrm{ht}(J^n)=\mathrm{ht}(J)$ for every $n>1$ so $\mathrm{ht}(J)=1$ iff…
user521337
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When is $\dim R_f = \dim R$?

Let $R$ be a commutative ring with $1$, and let $f \in R$ be an element of $R$ which is neither nilpotent, nor a unit (assuming there exists such an element in $R$). Let $R_f$ be the localization of $R$ with respect to the multiplicative subset $\{…
Malkoun
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Example: Krull dimension 1 but not a PID

It's easy to prove that if $A$ is a PID which is not a field then $\dim A= 1$. What is a counterexample to the converse? Thanks for any insight.
Mr. Chip
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Krull dimension of Noetherian local rings is finite

Does anyone know an "elementary" proof of the fact that a Noetherian local ring has finite Krull dimension? The one I know is from Atiyah&Macdonald's book Introduction to Commutative Algebra, where they use Hilbert functions (which is not an…
Diego
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Intuition on Krull Principal Ideal Theorem Proof's Major Ingredients

The krull principal ideal theorem I am interested in is the one involving that over Noetherian ring, minimal primes over the principal ideal must be height at most 1. In the proof, it involves symbolic power of prime ideals localization at the…
user45765
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Krull dimension of local Noetherian ring

Let $R$ be a commutative local Noetherian ring and $\mathfrak{m}$ its maximal ideal. Prove that, if $\mathfrak{m}$ is principal, then $\mathrm{dim}(R)\leq 1$ (the Krull dimension of the ring). Thank you.
Adrian Manea
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definition of Krull dimension of a module

Let $R$ be a commutative ring with $1$. We know that the Krull dimension of $R$ is by definition the length of the longest chain of prime ideals of $R$. Now if $M$ is a $R$-module, the Krull dimension of $M$ is by definition…
Leo
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Local ring of Krull dimension zero

In commutative rings text books it is usually asked to prove that as long as $(R,m)$ is a Noetherian local ring, the following are equivalent: (i) $m^n=m^{n+1}$ for some integer $n$; (ii) $m^n=0$ for some $n$; (iii) the Krull dimension of…
karparvar
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Krull dimension of quotient rings

This question is very related to this other question. I have an alternative solution to the ones proposed in the answers, and I'd like to know if it is correct. I want to find the dimension of $A=\mathbb{C}[x,y]/(x^3-y^2)$ using the dimension theory…
Daniel Robert-Nicoud
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Kodaira dimension and Canonical Ring

I have the following definition of the Kodaira dimension of a smooth variety $X$: $$k(X):= \begin{cases} \max \dim \phi_{|nK_{X}|}(X) & \exists n:|nk|\neq \emptyset for \\ -\infty & \text{otherwise} \end{cases}$$ Where $\phi_{|nK_{X}|}$ is the map…
STCJ
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Dimension of cone of projective variety

Let $X \subset \mathbb{P}^n$ be a nonempty projective variety. Show that the dimension of the cone $C(X):=\{0\} \cup \{(x_0,...,x_n)\in \mathbb{A}^{n+1}:(x_0:...:x_n)\in X\}$ is dim$X+1$. I know how to prove dim$C(X) \geq$ dim$X+1$: Let $\varnothing…
Chi Cheuk Tsang
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A Noetherian ring with Krull dimension one which is not a Dedekind domain

Can someone give me, with proof, an example of a Noetherian ring which has Krull dimension one but is not a Dedekind domain? I think it would also be instructive to see other "near misses."
Tony
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Krull dimension of local Noetherian ring (2)

Let $(A,\mathfrak{m})$ be a local Noetherian ring and $x \in \mathfrak{m}$. Prove that $\dim(A/xA) \geq \dim(A)-1$, with equality if $x$ is $A$-regular (i.e. multiplication with $x,$ as a map $A\rightarrow A$ is injective). The dimensions are…
Adrian Manea
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$\dim \mathbb K[x,y,z]/(xy,xz,yz)$

If $\mathbb K$ is a field I would like to find $$\dim \mathbb K[x,y,z]/(xy,xz,yz).$$ I'm starting to study the concept of dimension of rings and I don't know the basic tools and techniques to discover the dimensions of non-trivial rings like this…
user75086
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