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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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Definition of subintegral extensions

In his paper "On Seminormality", Swan defined subintegral extensions as follows: An extension of commutative rings $A\subseteq B$ is subintegral if: (1): $B$ is integral over $A$. (2): $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$ is bijective and…
LI Yuan
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Induced map of quotient modules is surjective

Let $R$ be a Noetherian domain, let $\mathfrak{m}$ be a max ideal and $\phi:P\to Q$ an $R$-map between finitely generated modules. Suppose that $\forall f\not\in \mathfrak{m}$, the map $\phi_f:P_f\to Q_f$ is not surjective (in particular $\phi$ is…
FunctionOfX
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Going-down theorem hypothesis

Something I don't get form the hypothesis of this theorem. If $C$ is the integral closure of $A$, and $A$ is integrally closed (since $A$ is integral domain, it's integrally closed over its field of fractions), then $A=C$. If $B$ is integral over…
Mand
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Some question about localization

Let $S$ be a graded ring generated by finite elements of $S_1$ as $S_0$-algebra and let $M$ be a graded $S$-module. For $m \in M$, if $m=0$ in $M_f$ for all generators $f \in S_1$, then $m=0$?
Sang Cheol Lee
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Must a local homomorphism from a Noetherian local ring to an artinian local ring factor through a power of its maximal ideal

Let $f : B\rightarrow A$ be a local homomorphism of Noetherian local rings where $A$ is moreover Artinian. Must $f$ factor through $B/m_B^n$ for some $n$?
stupid_question_bot
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Every prime power ideal in a Noetherian Ring of dimension one can be written uniquely as a power of a prime.

I am using Atiyah MacDonald to study commutative algebra and in Chapter 9, it says the following: If you look at the paragraph after the proof of 9.1, you can see that to get unique factorization, we must also have that given a nonzero prime ideal…
LoneStar
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Radical of ideal: Radical of $4\mathbb Z = 2\mathbb Z$?

The definition says that $x^n$ is in the ideal but I don't see what positive integer $4$ can be raised to in order to get $2$.
user719023
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Significance of the smallest $n \in \mathbb{N}$ such that $\sqrt{I}^n \subseteq I$?

Assuming we are in a Noetherian ring $R$ where such an integer always exists for a given ideal $I$, I am looking for any information on properties this integer possesses (in relation to $I$). Formally, let $n : = \text{min} \{k \in \mathbb{N} \mid…
beanshadow
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About factors of tensor products

Given $A,B,C$ Modules over a commutative ring $R$ when I can say that $A \otimes C \simeq B\otimes C\Rightarrow A\simeq B$?
Fabrizio
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Basic question on associated primes.

Suppose $\mathfrak{p},\mathfrak{q} \in \mathrm{Ass}(M)$ such that $\mathfrak{p}\subsetneq \mathfrak{q}$. Does there always exist $\mathfrak{p}'\in \mathrm{Ass}(M)$ such that $\mathfrak{p}\subsetneq \mathfrak{p}'\subset\mathfrak{q}$ and…
Jehu314
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Generalization of Gauss lemma to integrally closed integral domains

Let $R_1 \subset R_2$ be two integral domains. If $f \in R_1[x]$ is a polynomial which factors as $gh$ with $g,h \in R_2[x]$, then the coefficients of $g$ and $h$ are integral over $R_1$. I tried a bunch of stuff, but didn't really get anywhere…
katana_0
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"Dimension formula" for $K$-Algebra

There is this exercise in the book on commutative algebra I'm reading: Let $K$ be a field and $A$ an integral domain which is a finitely generated $K$-algebra. Let $\mathfrak a \subset A$ be an ideal and $$\mathrm{ht}(\mathfrak a):=\min_{\mathfrak p…
kade
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going between property

Let $R'\in R$ be an integral extension of rings with $R'$ a $K$-algebra finitely generated . Consider a chain of different prime ideals in $R'$ , $P_{1}\subsetneq P_{2}\subsetneq P_{3}$ Such that there are prime ideals in $R$ , $ Q_{1},Q_{3} $ …
Vegan Maths
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Computing the radical of an ideal using Nullstellensatz

How do you compute the radical of the ideal $\langle\,x^3+1\,\rangle$ in $\mathbb{C}[x]$? I know that you can use Nullstellensatz but I don't know.
qwertycheese7463
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A question about the uniqueness in Hilbert's Nullstellensatz

Suppose $k$ is an algebraically closed field. One of the formulations of Hilbert's weak Nullstellensatz is that each maximal ideal in $k[x_1, \dots , x_n]$ has the form $\langle x_1 -a_1 , \dots , x_n - a_n\rangle $, where $a = (a_1 , \dots , a_n)$…
Werner Germán Busch
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