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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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preservation of localness among certain Krull domains

Let $R$ be a local Krull domain, and let $\mathfrak p$ be a height one prime ideal whose class in the divisor class group is non-torsion. (That is, $\mathfrak p^{(n)}$ is non-principal for all $n$.) Obviously, this limits what $R$ can be -- e.g.…
calearner
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Powers of prime ideals

I was reading through Atiyah-MacDonald and they mention that if a ring $A$ is a Noetherian domain of dimension 1 has the property that every primary ideal is equal to the product of a prime ideal (i.e. if A is a Dedekind domain), then every ideal in…
user55600
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generators of a prime ideal in a noetherian ring

Suppose $R$ is a Noetherian ring and $P$ is a prime ideal. If the number of generators of $PR_P$ as an ideal in $R_P$ is $n$, can we say anything about the number of generators of $P$ as an ideal of $R$?
Zuben
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Every prime ideal of a finitely generated $\mathbb{R}$-algebra is an intersection of maximal ideals?

Why must every prime ideal of a finitely generated $\mathbb{R}$-algebra (e.g. $\mathbb{R}[X_1,X_2]$) be the intersection of the maximal ideals containing it? This doesn't follow from the version of the Nullstellensatz I've seen: if $k$ is an…
Harry Macpherson
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Is every ideal in $\hat{A}$ extended?

Let $A$ be a Noetherian ring, $I\subset A$ an ideal, $\hat{A}$ the $I$-adic completion. Is it true that every ideal of $\hat{A}$ is of the form $\hat{J}$ for some ideal $J\subset A$?
ashpool
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Prime ideal containing only zero divisors

Let $R$ be a Noetherian ring and $p\subset R$ be a prime ideal containing only zero divisors. Does it follow that $p$ is an associated prime of $R$? In other words is there some $x\in R$ such that $p=Ann(x)$?
Gazerun
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Ideals generated by irreducible elements

Let $R$ be a UFD and $f_1,\dots,f_n$ be irreducible elements of $R$. Does it follows that the ideal $\langle f_1,\dots,f_n\rangle$ is a prime ideal?. If it's not true in general then is it true in $k[x_1,\dots,x_n]$ where $k$ is an algebraically…
omar
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the reduced locus of a Noetherian ring

Let A be a Noetherian ring, Is the set of prime ideals $\{p\in \operatorname{Spec} A| A_p$ is a reduced ring $\}$ an open subset of $\operatorname{Spec} A$ in Zariski topology?
Yubin
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Some question ideal of variety

For an affine variety $X=V(x^{2}+y^{2}-1, x-1)$, I found the ideal of $X$, $I(X)=\langle x-1,y\rangle$. But I don't know $I(X)=\langle x^{2}+y^{2}-1, x-1\rangle$.
Sang Cheol Lee
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Krull dimension of tensor product

Let $f: (R,m) \rightarrow (S,n)$ be a morphism of local Noetherian rings. Let $M$ be a finite $R$-module and $N$ a finite $S$-module such that $\operatorname{Supp}M = \operatorname{Spec} R$ and $\operatorname{Supp}N = \operatorname{Spec}…
Manos
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Chain of prime ideals of maximal length

Consider the domain $R=\mathbb{C}[x,y]/(y^2-x^3)$. What would be an example of a chain of prime ideals of $R$ of maximal length?
Dave
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Proof of Hensel's lemma

I am reading up the proof of Hensel's lemma here. On page 2, after equation 2, the author concludes that the degree of $\delta h_k$ is less than $n$ since the degree of $\Delta$ and $\epsilon g_k$ is less than $n$. I am not sure I understand this.…
BMI
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Prime ideals not containing the conductor of a finite extension of a domain

Let $A\subseteq B$ be integral domains, where $B$ is contained in the quotient field $K$ of $A$, and $B$ is finitely generated as an $A$-module. Let $\mathfrak{f}=\{a\in A\mid aB\subseteq A\}$ denote the conductor. (As is well known, it is an ideal…
Matthé van der Lee
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Is the support of an Artinian module finite?

If $R$ is an Artinian ring then it has finite maximal ideals. If $M$ is an $R$-module Artinian. ($R$ be a commutative Noetherian ring). Then, is $Supp(M)$ finite? Thanks.
Tri Nguyen
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Is the infinite intersection of prime ideals also a prime ideal?

I'm currently reading the proof of the fact that the intersection of all the prime ideals of the commutative ring $A$ is nilradical of $A$. The proof takes every non-nilpotent element $f$, and proves that there exists a prime ideal which does not…
user67803
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