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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

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Height of a prime ideal and number of generators of its localization

This question is very related to this one: generators of a prime ideal in a noetherian ring. Let $\mathfrak{p}$ be a prime ideal in a Noetherian ring and let $k$ be its height. Further suppose that $f_{1},\dots, f_{k} \in \mathfrak{p}$ generate the…
Sebastian
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Maximal element in set of Ann(m) for m in M is prime

Exercise 15.1.32 in Dummit & Foote, along with the included hint, is Suppose that $M$ is a $R$-module and that $P$ is a maximal element in the collection of ideals of the form $\operatorname{Ann}(m)$, for $m\in M$. Prove that $P$ is a prime…
Avi Steiner
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Question about completions.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. I have some questions about Corollary 10.3 and Corollary 10.4. Why the sequence $$ 0 \to \frac{G'}{G' \cap G_n} \to \frac{G}{G_n} \to \frac{G''}{pG_n} \to 0 $$ is…
LJR
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Question about the lying over theorem.

I have some questions about the proof of the Lying over theorem in the book Introduction to commutative algebra by Atiyah and Macdonald. (1) In the proof of Theorem 5.10 of Page 62, is the map $\alpha: A \to A_{\mathfrak{p}}$ the natural map which…
LJR
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Waring’s problem in commutative rings

Let $k\geq 2$ be a fixed integer. If $R$ is a commutative, integral, unital ring, the Waring height of an element $r\in R$ is the smallest number of $k$-ths powers whose sum is $r$ (this height can equal $\infty$, when $r$ cannot be written as a sum…
Ewan Delanoy
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Is the completion of $A[x]$ the same as $\widehat{A}[[x]]$?

Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$. Is $\widehat{A}[[x]]$ the completion of $A[x]$ at $(\mathfrak{m},x)$? We certainly have surjections $$\widehat{A}[[x]]/(\mathfrak{m}^n\widehat{A} + x^n\widehat{A})\cong…
stupid_question_bot
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Proposition 4.11 in Eisenbud's Commutative Algebra

(Proposition 4.11 and its proof are attached below.) In the proof, I'm not sure why $S[\alpha_1]=S[x]/(g)$. As I understand it, we can define the surjective map $S[x]\to S[\alpha_1]$ by sending $x$ to $\alpha_1$. But couldn't the kernel of this map…
klein4
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Atiyah Ex5.29: Local ring of a valuation ring

Let $A$ be a valuation ring of a field $K$. Show that every subring of $K$ which contains $A$ is a local ring of $A$. This problem is already asked and answered at mathoverflow. But I can't understand why $PA_P \subset M_B$ at step (b) of the…
Gobi
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Is $\mathbb Q[[x]]$ compact with the $(x)$-adic topology?

I'd like to understand what the $(x)$-adic topology looks like here. I know that $\mathbb Q[[x]]$ is the completion of $\mathbb Q[x]$. I feel like this should not be compact. To show its not compact we want to find some open cover which cannot be…
Mark Murray
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special case of Nagata's Lemma (Matsumura p.86)

Let $K$ be a field and $R$ a valuation ring of $K$ with maximal ideal $m_R$. Let $a \in R$ such that $1-a \in m_R$. Statement: For any $s$ that is not a multiple of the characteristic of $R/m_R$, the element $(1+a+a^2+\cdots+a^{s-1})^{-1}$ is…
Manos
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Elementary question about integral extensions

I'm reading page $59$ of Reid's "Undergraduate commutative algebra" book. In example (ii) it says, $k[x^{2}] \subset k[x]$ is an integral extension. How do we know this? I mean, in order to show this we must take a polynomial $f(x) \in k[x]$ and…
user6495
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Reduction of ideals in a commutative ring

Is it possible to have an infinitely generated reduction of a finitely generated ideal in a commutative ring with identity ? If yes, why ? If no, an example to this effect will be helpful. Thank you. Edit. By reduction, I meant the following: if…
user62198
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Ring With Infinitely Many Ideals but Proper has only Finitely Many

I am looking for a commutative ring R with identity, with infinitely many ideals, yet every proper ideal contains (as a subset) only finitely many ideals of R. I know such a ring cannot be Artin (because Artin rings have only a finite number of max…
Wdunn
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Flatness versus vanishing of Tor-groups for a non-finitely generated module

This is something I should probably know, but it is escaping me at the moment. Let $A$ be a commutative noetherian ring. The following corollary of Nakayama's lemma is well-known (for instance, this is Atiyah-Macdonald exercises 7.15 and 7.16) If…
tkr
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Going down theorem

Here $A$ is a commutative ring with unity. How to show that going down theorem holds for $A$ contained in $A[x]$, the polynomial ring. Lying over is ok. I cannot do the other part.
kushal
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