# 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.

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### Krull dimension of localization of $R=k[x_1,\dots,x_p]/\langle (x_i x_j)_{1\leq i<j\leq p}\rangle$ at maximal ideal.

Let $k$ be a field. Consider the ring $R=k[x_1,\dots,x_p]/\langle (x_i x_j)_{1\leq i<j\leq p}\rangle$. Let $A$ be its localisation at the maximal ideal $m=\langle x_1,\dots,x_p\rangle$. Why $A$ has ...
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### Krull dimension of localized module [closed]

Let $\frak p$ be a prime ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. Is it true that $\dim_{R_{\frak p}}M_{\frak{p}}+\dim_RM/{\frak p}M\leq \dim_RM$? (Note that ...
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### Property related to the Krull dimension of modules which form an exact sequence

This is a claim from matsumura's commutative ring theory given in the proof of dimension theorem, and I confused about it for a while: Given a finite generated module $M,M',M''$over noetherian ring $R$...
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### A ring with Krull dimension 2 and principal ideal proper factor.

Is there an example of a ring with Krull dimension 2 where $R/I$ is a principal ideal ring for every nonzero ideal $I$? Is the polynomial ring $\Bbb Z[x]$ the answer?
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### Height of irreducible polynomials

I started to read about dimension of varieties. The encountered the algebraic version of it, Krull Dimension. To get intuition I am trying to calculate heights of primes. I found that Atiyah Macdonald ...
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### Krull Dimension is defined by induction on ordinals.

I was reading the book "Serial Rings" by Gennadi Puninski, there it is written, "The Krull dimension $Kdim(M)$ of a module $M$ is defined by induction on ordinals". I can't ...
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### Inequality of the dimension of the formal power series ring when the ring is non-noetherian

We generally know that if $R$ is Noetherian then we have that $\dim (R[[X]])=\dim(R)+1$. If we drop the assumption that $R$ is Noetherian does the inequality $\dim (R[[x]]) \geq \dim(R)+1$ hold true?...
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### 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 ...
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### Krull dimension of $K[[x]]$ [duplicate]

Well, I know that this is noetherian but I don't really understand how I can get a chain of prime ideals here to find the Krull dimension and assure that it is the supremum.
### Describe prime ideals and Krull dimension of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$
I want to describe the prime ideals of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$, where $\overline{\mathbb{Q}}$ denotes the integral closure of $\mathbb{Q}$ in $\mathbb{C}$, ...