# Tagged Questions

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### Principal ideal domain is universally catenary …

... actually, even more general statement is true: Theorem. Every regular ring is universally catenary. (see for example Algebraic Geometry by Qing Liu, Corollary 2.16, Chapter 8) Though, the ...
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### Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivlance does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
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### Proof details of Theorem 11.1 in Atiyah-Macdonald

I have some trouble filling in the details of this proof from Atiyah-Macdonald. In this result, the authors assume what follows: 1) $A = \oplus_{n=0}^\infty A_n$ is a Noetherian graded ring, and ...
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### Transcendence degree of $K[X_1,X_2,\ldots,X_n]$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every ...
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### Krull dimension of the injective hull of residue field

Let $(R,\mathfrak{m})$ be a noetherian local ring, and $E=E_R(R/\mathfrak{m})$ the injective hull of $R/\mathfrak{m}$. What do we know about the Krull dimension of $E$? Thank you.
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### Prime ideals of height less than the dimension

Let $A$ be a noetherian local ring with maximal ideal $\mathfrak{m}$ of height $d$, and suppose $\mathfrak{p}_1, \ldots, \mathfrak{p}_s$ are prime ideals of height $i - 1 < d$. It's quite clear ...
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### Which definition of dimension came first?

In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, ...
Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...