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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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A problem about Noetherian local ring

This is a problem I quote from P107 of Matsumura's book Commutative ring theory. Let $(A,m)$ be a Noetherian local ring. $q$ is a $m$-primary ideal. Then we have $l(q^n/q^nm)l(A/q)\geq l(q^n/q^{n+1})$, where $l$ means the length of module. The book…
Intoks Liobein
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Length of a Module under Completion of the Ring

My question refers to following conclusion in Liu's "Algebraic Geometry" at page 290 (Thm 4.16): Why does $e'_x= length_{\hat{B}}(W_{\hat{B}/\hat{A}}/\hat{B})$ hold? Here $e'_x$ is introduced as $e'_x = length_{\mathcal{O}_{X,x}}(\omega_f/…
user267839
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$\dim R = n-1$ if and only if $ R\cong k[x_1,\dots,x_n]/(f) $

Let $k$ be a field and $R$ a $k$-algebra generated by $ n<\infty $ elements. Show that the following are equivalent: (i) $R$ is a domain of dimension $ n-1 $ (ii) $ R\cong k[x_1,\dots,x_n]/(f) $ where $ k[x_1,\dots,x_n] $ is a polynomial ring and…
user346096
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Primary decomposition and contraction

Let $R$ be a Noetherian ring, $Q$ a $P$-primary ideal in $R$ for some $P\in \operatorname{Spec}(R)$ and $S$ an arbitrary multiplicative set in $R$ which does not meet $Q$, i.e. $Q\cap S=\varnothing$. What can be said of the contraction:…
walkar
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Given an arbitrary commutative ring $A$, does integral closure always exist?

As for an arbitrary field $K$, we know that its algebraic closure always exists and it is unique up to an isomorphism. However, when we talk about integral closure of some commutative ring $A$, we are always given $A$ as some subring of a larger…
Sam
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Is there a name for the intersection of the maximal ideals containing an ideal?

Clearly it's a prime ideal containing the radical. Do you just call it the "Jacobson radical of I?"
Daniel McLaury
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Local homomorphism and $m$-primary ideal

Let $\phi:R \longrightarrow S$ be a local homomorphism of local rings. Let $m_R$ and $m_S$ denote the unique maximal ideals of the local rings $R$ and $S$ respectively. Under what conditions can we say that $\phi(m_R)S$ is $m_S$-primary? Is there a…
Sam
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Prime and maximal ideals in finite algebras embedding

I assume each algebra here is commutative with a unit. Let $\nu : A \hookrightarrow B$ be a finite embedding of algebras (that is $B$ is finitely-generated as an $A$-module). I want to show the following: That the map $\nu^{-1} : \mathrm{Spec}(B)…
LinAlgMan
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Suppose that B is a finitely generated A-algebra. If B is a noetherian ring, is A noetherian?

I know how to prove that A noetherian implies B noetherian using Hilbert's basis theorem. However, I wasn't able to produce an answer to the converse which probably is false.
B. Rivas
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Irreducible polynomial over Dedekind domain remains irreducible in field of fractions

Let $\mathcal{O}$ be a Dedekind domain, $K$ its field of fractions. Suppose $f\in \mathcal{O}[X]$ is irreducible. Is it irreducible in $K[X]$? The motivation for my question is that this is true for UFD's, so it is natural to ask if it is still…
user46225
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Is there a DVR of char $0$ with non-perfect residue field?

I am trying to prove a claim in curves, and the example below shall contradict it. Is there a DVR, $(A,\mathfrak{m})$, of char $0$ such that the residue field $\mathcal{k}(A) = A/\mathfrak{m}$ is a non-perfect field? Basically I would like to…
Grobber
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Counter-example: Radical of a k-algebra

I have the following question. Let $k$ a field, $A$ a $k$-algebra, and $k\hookrightarrow K$ a field extension. It´s well known that $$Rad(A)\otimes_{k}K\subset Rad(A\otimes_{k}K)$$ ($Rad(A)$=radical of $A$). When is it $Rad(A)\otimes_{k}K=…
user320224
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Existence of a minimal primary decomposition

Let $R$ be a commutative ring with unity and $I$ be a decomposable ideal of $R$. Question 1 If $Q$ is a primary ideal containing $I$, then does there exist a minimal primary decomposition of $I$ which has $Q$ as its element? Question 2 Let…
Rubertos
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Atiyah–Macdonald exercise 3.14

I am trying to understand the hint in exercise 3.14 in Atiyah–Macdonald. Let $M$ be an $A$-module and $\mathfrak a$ be an ideal of $A$. Suppose that $M_{\mathfrak m} = 0$ for all maximal ideals $\mathfrak m \supseteq \mathfrak a$. Prove that $M =…
Earthliŋ
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If $S$ is a Noetherian graded ring, $S_{(f)}$ is also Noetherian?

Let $S = \sum_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $> 0$. Let $S_{(f)}$ be the degree $0$ part of $S_f$. If $S$ is Noetherian, $S_{(f)}$ is also Noetherian?
Makoto Kato
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