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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.

16857 questions
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Isomorphism between two localizations

I am doing exercise 3.23 in Atiyah Macdonald and in the first part of the problem they ask to show that the ring $A_f = S^{-1}A$ where $S = \{1,f,f^2 \ldots \}$ depends only on the choice of the basic open set $X_f$ and not on $f$. For…
user38268
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If $P$ is a prime ideal in a commutative Noetherian local ring $R$, is $P\hat{R}$ a prime ideal in $\hat{R}$?

Do prime ideals expand to prime ideals in the completion? I believe this is the case since I think $R/P\equiv \hat{R}/P\hat{R}$, although Atiyah-Macdonald explicitly mentions the preservation of quotients only with respect to powers of maximal…
Jake Voigt
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primary ideal of regular local ring

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d$. Let $P$ be a prime ideal of height $d-1$. I want to know if $P^2$ is always a $P$ primary ideal ie if $P/P^2$ is torsion free as $R/P$ module. Thanks. Let me explain few of my…
A.G
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Tensor product of modules over quotients by annihilators

If M and N are modules over some commutative ring A and $\mathfrak{a} \subset \operatorname{Ann(M)} \cap \operatorname{Ann(N)}$ is an ideal, is it true that $M \otimes_A N \cong M \otimes_{A/\mathfrak{a}} N$ as A-modules? I think that I can…
Paul
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If every prime ideal is a contracted ideal, does it imply that the induced map between spectrums is surjective?

Let $f: A \rightarrow B$ be a ring homomorphism between commutative rings with identity. Then there exists an induced map $f' : Spec(B) \rightarrow Spec(A)$. If $f'$ is surjective, then clearly every prime ideal of $A$ is a contracted ideal. Now my…
Maria
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Principal ideals in completions of Zariski rings

Let $A$ be a Noetherian ring and $\mathfrak{a}$ some ideal contained in the Jacobson radical of $A$. Now $A$ is endowed with the $\mathfrak{a}$-adic topology, i.e. $A$ is a Zariski ring. If $\mathfrak{b} \subset A$ is an ideal so that $\mathfrak{b}…
Paul
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Is Serre's $S_1$ condition equivalent to having no embedded primes?

Today I tried to prove that if a Noetherian ring $A$ satisfies Serre's $R_0$ and $S_1$ conditions, then $A$ is reduced. Now we recall that $R_0$ means the localization at any minimal prime is a field (Nakayama's lemma) and $S_1$ means that…
David Benjamin Lim
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Question on proof from Eisenbud's Commutative Algebra.

I don't understand the second part of the proof of Corollary 4.8 (Nakayama's Lemma) in Eisenbud's Commutative Algebra. Let $I$ be an ideal contained in the Jacobson radical of a ring $R$, and let $M$ be a finitely generated $R$-module. (a) If…
nknx
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Localizing a localization at a prime ideal

When I did some reading on localization, I stumbled upon a thought which I'm not able to prove. This might be right or wrong in a trivial fashion, but I'm stuck nonetheless. Let A be an integral domain and suppose $S \subset A$ is multiplicatively…
tommy
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Classification of radical ideals of $\mathbb{Z}[X]$

The prime ideals of $\mathbb{Z}[X]$ are $0$ $(p)$ for prime integer $p$ $(f)$ for irreducible polynomial $f$ $(p, f)$ for prime integer $p$ and irreducible polynomial $f$ that remains irreducible mod $p$. The radical ideals are intersections…
Tim
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Atiyah and Macdonald Exercise 1.27

Please do not ruin the fun by telling me why $\mu$ is surjective! I am having trouble understanding the idea of the coordinate functions on the affine algebraic variety $X$. I am trying to understand that $P(X)$ is generated as a $k$-algebra by the…
Daniel
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Can we have a Primary Avoidance Theorem ?

Prime Avoidance Theorem says: Let $ P_1, P_2,\dots, P_n $ be prime ideals in a commutative ring $R$ and let $I$ be an ideal of $R$ such that $ I \subseteq P_1 \cup P_2 \cup \cdots \cup P_n$. Then $ I \subseteq P_k $ for some…
user82664
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Is there a 'Noetherialization' Process for Rings (Can I take a ring and get a "similar" Noetherian one)

I had asked this question in one of the seminars I attend and they said I should ask it here. Let R be a commutative ring with unity where $1 \neq 0$ and suppose that R is not Noetherian (i.e. all of the 3 equivalent conditions: max.c, acc, and…
Countable
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Zero divisors and height of prime ideals in Noetherian rings.

Let $R$ be a noetherian ring, $x\in R $ be a non zero divisor, and $P$ a prime ideal of $R$ which is minimal over $(x)$. I'm trying to show that $\operatorname{ht}P=1$. Also if $Q$ is a prime ideal of height $1$, is there a non zero divisor in…
i.a.m
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Does the residue field and the fraction field uniquely determine a complete DVR?

Assume we have two complete DVRs $R_1$, $R_2$ with finite residue fields. Assume their residue fields and fraction fields are isomorphic as abstract fields. Can $R_1$ and $R_2$ be non-isomorphic (as unital rings)? EDIT: from Serre's textbook on…
user692020
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