Mathematics Stack Exchange
Mathematics Stack Exchange

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.

374 questions
6
votes
3 answers

Why is $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ a Dedekind domain?

What is the best way to understand that $D:=\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is a Dedekind domain? I first noticed that $X^2+Y^2-1$ is irreducible in $\mathbb{Q}[X,Y]$ since it is $Y-1$ Eisenstein in $\mathbb{Q}[Y][X]$. It follows that…
Tiffany Hwang
  • 1,179
  • 9
  • 19
6
votes
1 answer

Krull dimension of $\mathbb{Z}[x_1,\dots,x_n]$

I'm trying to prove that the Krull dimension of $\mathbb{Z}[x_1,\dots, x_n]$ is $n+1$. I know there is a result that says $$\dim(A[x_1,\dots, x_n])=n+\dim(A),$$ when $A$ is a Noetherian ring, but I was outlined a proof of this by another method…
Chris Birkbeck
  • 1,416
  • 9
  • 19
6
votes
1 answer

Finite dimensional local rings with infinitely many minimal prime ideals

Is there a finite dimensional local ring with infinitely many minimal prime ideals? Equivalent formulation: Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of minimal prime sub-ideals of $\mathfrak p$ is…
Pierre-Yves Gaillard
  • 18,672
  • 3
  • 44
  • 102
6
votes
0 answers

Krull dimension of the adele ring

Let $k$ be a number field and $\mathbf{A}_k$ the adele ring of $k$. What can be said about the Krull dimension of $\mathbf{A}_k$? More generally, I do not know if something can be said about the Krull dimension of an infinite product of rings: is it…
Gaussian
  • 382
  • 1
  • 9
6
votes
2 answers

$K[X^2,X^3]\subset K[X]$ is a Noetherian domain and all its prime ideals are maximal

Consider $K$ field and consider the ring $R=K[X^2,X^3]\subset K[X]$. It is clear that $R$ is not a Dedekind domain, since with the element $X$ one immediately see that it is not integrally closed. But $R$ is a Noetherian domain and every non trivial…
Lei Feima
  • 181
  • 7
6
votes
2 answers

Is there a non-artinian noetherian ring whose non-units are zero-divisors?

Is there a non-artinian noetherian ring whose non-units are zero-divisors? Equivalent formulation: Is there a noetherian ring of positive dimension whose non-units are zero-divisors? [In this post, "ring" means "commutative ring with one", and…
Pierre-Yves Gaillard
  • 18,672
  • 3
  • 44
  • 102
6
votes
1 answer

Dimension of a product of projective varieties

Maybe this is a dumb question, but better safe than sorry. In Hartshorne, Exercise I.3.15 and I.3.16 we are asked to examine products of affine and projective varieties. I.3.15 has been smooth sailing. The reader is asked to prove: a) $X \times Y$…
A. Thomas Yerger
  • 16,021
  • 2
  • 35
  • 69
6
votes
2 answers

Final step of exercise 11.7 from Atiyah-Macdonald ($\dim A[x]=\dim A+1$)

Ex. 11.7 from Atiyah-Macdonald is basically to prove $\dim A[x]=\dim A+1$ for $A$ noetherian. From exercise 11.6, we get $\dim A[x]\geq\dim A+1$, so we are left to prove "$\leq$". I've followed the hint and proved that $\text{ht}(p[x])=\text{ht}(p)$…
rmdmc89
  • 8,785
  • 2
  • 18
  • 52
6
votes
2 answers

Example of a commutative noetherian ring with $1$ which is neither domain nor local and has a principal prime ideal of height $1.$

I am trying to construct an example of a ring satisfying the followings. A commutative noetherian ring with $1$ which is neither domain nor local and has a principal prime ideal of height $1.$ I know that a local noetherian ring having a height…
user371231
  • 2,251
  • 1
  • 8
  • 20
6
votes
0 answers

When will $A[x_1, \ldots, x_n]$ satisfy the dimension formula?

What property should $A$ satisfy so that $A[x_1, \ldots, x_n]$ satisfies the dimension formula, $$\mathrm{dim}(A[x_1, \ldots, x_n]) = \mathrm{dim}(A[x_1, \ldots, x_n]/\mathfrak{p}) + \mathrm{ht}(\mathfrak{p}),$$ for any prime ideal…
Zoey
  • 533
  • 3
  • 12
6
votes
2 answers

What is the Krull dimension of $\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]$?

I am studying commutative algebra and saw the following question in one of the tests: What is the Krull dimension of $R=\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]?$ I know that $\dim R \leq \dim\mathbb{Q}[x,\ y,\ z]=3$ and $\dim…
Tommy1234
  • 511
  • 2
  • 12
6
votes
1 answer

Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
Not Buying It
  • 461
  • 2
  • 7
6
votes
1 answer

Depth of a module over local ring and vanishing of Ext functor

I'm studying depth of $A$-modules, where $A$ is a noetherian ring, in Matsumura's Commutative Algebra text and I'm experiencing some trouble understanding the proof of a basic result. I think all of you do know the following fact: $$\mathrm{depth}_I…
Caligula
  • 2,077
  • 7
  • 16
5
votes
2 answers

Calculation of codimension

I've just learnt the idea of codimension, then I try to find some exercises to calculate the codimension of some maximal ideal $J$ in the ring $R=k[x,y,z]/I$, I find it somewhat tricky. For example, If $R=k[x,y,z]/(x^2y^2z^2,xy^3z^2)$ and…
Bamqf
  • 399
  • 1
  • 5
5
votes
2 answers

surjective map of rings with same dimension

Let $A \to B$ be a surjective homomorphism between (unital) noetherian commutative rings with the same Krull dimension. Is the kernel of this map nilpotent ? Thanks to Makoto Kato and Martin Brandenburg, it seems that the answer to the question is…
user65490
  • 295
  • 1
  • 10
1 2
3
…
24 25