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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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Primary ideal exercise

I have an exercise about the properties of primary ideal. It's Exercise 15.17 of "Step in commutative algebra", R. Y. Sharp. Let $(A,\mathfrak{m})$ be a local ring and $I$ be a proper ideal of $A$. Prove that the following statements are…
Rachel
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Question about the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$.

Given the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$. I want to compute $V(I)$, which is the intersection of all ideals containing $I$. And I also want to prove that $I$ can't be generated by two elements over $\mathbb C[x,y,z]$. How can I…
adrija
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Deducing a presentation for a complete intersection

Today, while reading some articles, I had this doubt trying to justify a passage: Hypothesis Suppose $O_K$ is some complete discrete valuation ring (it is the ring of integers of some complete field in my case, but it shouldn't be important). Let…
Warner
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Surjective homomorphism on Laurent polynomial ring, part II

This question is similar to the question link, with a stronger hypothesis. Let $A= \mathbb C [t^2,t^{-2}]$ and $B= \mathbb C [t,t^{-1}]$. Consider $f\in B$ with the form $f=(t-a_1)(t-a_2)\cdots(t-a_k)$ where $a_i\in \mathbb C\setminus \{0\}$ with…
Binai
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Surjective homomorphism in Laurent polynomial ring.

Let $A= \mathbb C [t^2,t^{-2}]$ and $B= \mathbb C [t,t^{-1}]$. Consider $f\in B$ with the form $f=(t-a_1)(t-a_2)\cdots(t-a_k)$ where $a_i\in \mathbb C\setminus \{0\}$ and let $I$ be the ideal generated by $f$ in $B$. Define the map $\phi: A \to B/I$…
Binai
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Is an ideal the intersection of contractions of expansions to localizations at its minimal primes

Let $I$ be an ideal in a Noetherian ring $R$. Is $I=\cap(IR_P\cap R)$ where the intersection is taken over all minimal primes of $I$? If not, is it true if we assume $I$ has no embedded primes? I am motivated to ask this because the statement is…
Gene Simmons
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Height one prime avoidance in normal domains

Let $R$ be a Noetherian normal domain. Let $X$ be the set of height one prime ideals of $R$, and let $\mathfrak p \in X$. Can one have $$ \mathfrak p \subseteq \bigcup_{\mathfrak q \in X \setminus \{\mathfrak p\}} \mathfrak q? $$ Moreover, if…
calearner
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Definition of degree of commutative ring $d(A) $ based on Hilbert polynomial

I'm studying chapter 11 (Dimension Theory) in Atiyah / Macdonald - Intro to Commutative Algebra. Let $ A $ be a Noetherian local ring with $\mathfrak{m}$-primary ideal $\mathfrak{q}$. The book defines $d (A)$ as the common degree of the…
PeterM
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Going Down Theorem, AM

I'm trying to understand the proof of the going down theorem in Introduction to Commutative Algebra by Atiyah and Macdonald. My main confusion is when they say it suffices to show that $B_{\mathfrak q_1}\mathfrak p_2\cap A = \mathfrak p_2$. Isn't it…
kfriend
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Lemma 5.3.6 in Bruns and Herzog, Cohen-Macaulay Rings

In the picture we discuss the Stanley-Reisner ring over a simplicial complex $\Delta$. I do not understand the steps "(i) implies" and "(ii) implies", maybe I do not catch how to translate the complexes language into the element language. For…
Strongart
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Common regular sequence of ring and module

Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M\neq0$ a finite $A$-module. Suppose $$d=\min\{{\operatorname{depth}A,\operatorname{depth}M\}}\geq1.$$ Then does there always exists $a_1,\ldots,a_d\in\mathfrak{m}$ which is both an $A$-sequence…
ashpool
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What is $\overline{\{ x \}}$ in Atiyah-Macdonald?

On pg. 13 of Atiyah-Macdonald's "Introduction to Commutative Algebra": 18.ii) Prove that $\overline{\{ x \}}=V(p_x)$ What is $\overline{\{ x \}}$? Is it the closure of prime ideal $x$? I assumed it was so, and landed up in a lot of weird…
freebird
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How do we glue splittings together?

Let $M$ be a finitely-generated module over a Dedekind domain $R$. I need to show that $M = M_1 \oplus M_2$ where $M_1$ is torsion and $M_2$ is projective. It's clear we can do this locally: indeed, for any $\mathfrak{p} \in \operatorname{Spec} R$,…
Zhen Lin
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Associated primes of quotient module

Let $R$ be a Noetherian local ring of Krull dimension $d$, $M$ a finitely generated module over $R$. Suppose $\dim M=d$ and $K$ is a submodule of $M$ maximal with respect to the property that $\dim K\leq d-1$, then can we deduce that…
nick
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A nonexample of a prime submodule?

Let $M$ be an $A$-module. A submodule $P\subset M$ is called a prime submodule if it is proper and $am\in P$ implies $aM\subset P$ or $m\in P$. It is easy to see that if $P\subset M$ is a prime submodule, then the ideal $$(P:M):=\{a\in A \mid…
ashpool
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