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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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Containment of primary ideals

Suppose, $R$ is a noetherian ring. Let $P$ be a prime ideal in $R$. Let $Q$ be a $P$-primary ideal that contains $P^n$. Then does $Q$ contain $P^{(n)}$ which is the $n$th symbolic power of $P$ and is the $P$-primary component that occurs in any…
BMI
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Is $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$ noetherian?

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Z}_p$ be the $p$-adic completion of $\mathbb{Z}$. Similarly for $\mathbb{Z}_q$. Is $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$ noetherian?
Makoto Kato
  • 42,602
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Structure of the completion of $\mathbb{Z}[x]$ with respect to a maximal ideal

Can we explicitely describe the completion $R$ of $\mathbb{Z}[x]$ with respect to a maximal ideal $\mathfrak{m}\subset \mathbb{Z}[x]$? $(R,n)$ is a complete regular local ring of dimension two with mixed characteritic. I recently learned that that…
DonD
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About $\operatorname{Supp}_A M_{\mathfrak{p}}$

Let $M$ be a finitely generated $A$-module and $\mathfrak{p}\subset A$ a prime ideal. Is it true that $\operatorname{Supp}_A M_{\mathfrak{p}}$ is the closure of $\operatorname{Supp}_{A_{\mathfrak{p}}} M_{\mathfrak{p}} \subset \operatorname{Spec}A$ ?
ashpool
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does a prime in an extensions of integral domains remain radical?

Let $R\subset R'$ be an extension of integral domains. So we have an inclusion map $i:R\hookrightarrow R'$. Let $\mathfrak{p}\subset R$ be a prime ideal. We know that $\mathfrak{p}^e$ (generated by the image of $\mathfrak{p}$ under $i$) need not be…
adrido
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Is $S(E\otimes_AB)\cong S(E)\otimes_AB$?

Suppose A is a commutative ring, $E$ is an $A$-module, $B$ is an $A$-algebra, ${S}$ is the symmetric $A$-algebra functor. Is $S(E\otimes_AB)\cong S(E)\otimes_AB$? I try to use universial property, where ${S}$ is the left adjoint of the forgetful…
user93417
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Henselian rings with the same quotient field

I was recently reading these notes, where it is proved (a theorem of Kaplansky-Schilling) that a field that admits two distinct valuations with respect to which it is henselian is separably closed. A henselian field is the same thing as the data of…
Akhil Mathew
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Determine the kernel of a surjective morphism of differential modules

I am trying to find out how to complete the proof of the following statement. Let $R, S, R', S'$ be commutative rings and $(\psi, \varphi) : (R, S) \to (R', S')$ be a morphism from the algebra $\alpha : R \to S$ to $\beta : R' \to S'$, i.e. $\psi :…
Filip Chindea
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An integral ring extension that $B/M$ is not separable over $A/m$

I am trying to find an example like this. Let $A$ be integrally closed in its field of fraction $K$ and $L$ a finite Galois extension. Let $B$ be the integral closure of $A$ in $L$. Is there a constellation such that there a maximal ideal $m$ of…
Peter Patzt
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A Noetherian module annihilated by a power of maximal ideal must has finite length.

Let $M$ be a Noetherian $R$-module and $P^kM=0$ from some maximal ideal $P$ of $R$ and some integer $k$. How to show that $M$ has finite length? The length of a module is defined to be the maximum length of the chain of…
hxhxhx88
  • 5,257
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Localization of prime ideals

Let $A$ be a commutative ring with $1$. Suppose that $P \subseteq Q$ are prime ideals in $A$ and that $M$ is an $A$-module. Prove that the localization of the $A$-module $M_{Q}$ at $P$ is the localization $M_{P}$, i.e $(M_{Q})_{P} = M_{P}$. Hint…
user6495
  • 3,957
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Transitivity of norm in the general case

I'm looking for a reference for the following result (all rings commutative with unit) : Let $A$ be a ring, $B$ be an $A$-algebra which is free of finite rank as an $A$-module, $M$ be a free $B$-module of finite rank and $u$ an endormorphism of…
Zorba le Grec
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Fibre product is Noetherian

Let $A$ and $B$ be Noetherian rings and $f: A \rightarrow C$ and $g: B \rightarrow C$ ring homomorphisms. If both $f$ and $g$ are surjective show $\{(a,b) \in A \times B: f(a)=g(b)\}$ is a Noetherian ring. Here's a hint: show first (I already did…
user6495
  • 3,957
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does tensoring commute with taking images without flatness?

This is a follow-up to this question: was flatness really used in this argument? (Matsumura, Theorem 7.2). The author of the given answer says we need flatness to ensure that tensoring commutes with taking images. I still find this statement…
Manos
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$a^n=0\,\Rightarrow\, X-a\mid X^n$ (nilpotent Factor Theorem)

Proposition 1.6 in A. Kleiman A Term of Commutative Algebra says that if $R$ is a ring and $a \in R$, then the map $\pi:R[X]\rightarrow R$ given by $\pi(X) = a$ has kernel generated by $(X-a)$ and $R[X]/(X-a)$ is isomorphic to $R$. But if $a$ is…
rwilsker
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