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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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How to determine a primary decomposition of $(X^aY^b,(X+Y+Z)^c)$ in $k[X,Y,Z]$

I am trying to prove that the primary decomposition of $(X^aY^b,(X+Y+Z)^c)$ in $k[X,Y,Z]$, for a,b,c positive integers, is $(X^a,(X+Y+Z)^c) \cap (Y^b,(X+Y+Z)^c)$. The equality of the ideal and the intersection is easy to see, but I am not succeeding…
user198182
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Finding conditions for $\mathbb Z[i][X,Y]/(Y^2 - aX)$, $a \in \mathbb Z[i]$ to be regular

I am trying to find the dimension and the necessary and sufficient conditions under which $A[X,Y]/(Y^2 - aX)$ is regular, that is, the localizations of $A[X,Y]/(Y^2 - aX)$ at all maximal ideals are regular, where $A$ are the Gaussian integers and…
baltazar
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Atiyah-Macdonald p.108

I don't understand the following lines on p.108 (chapter 10) in Atiyah-Macdonald: Since we have a natural homomorphism $f:A\to \hat{A}$ we can regard $\hat{A}$ as an $A$-algebra and so for any $A$-module $M$ we can form an $\hat{A}$-module…
Rungo
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Find shortest primary decomposition.

Let $A=k[x,y,z]$ and let $T_1=(x,y)$, $T_2=(x,z)$. Define $I=T_1T_2$ and calculate the shortest primary decomposition of $I$. I dont know where to start and I am looking for hints, how should I think when I want to find a primary decomposition? I…
user117449
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Question on rank function.

In a previous question I asked about the fiber $M(P)=M_P / PM_P$ where $M$ is an $A$-module and $P$ a prime ideal of $A$. Later I introduced the rank function $$rk_M : \text{Spec} A \to \mathbb{N} \cup \{ \infty \}$$ given by $$P \mapsto \dim_{A(P)}…
user117449
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How to show a given algebra is not generated by one element

Suppose I have a $k$-algebra $k[x,y]/\langle f\rangle$, where $f$ is a (given, fixed) irreducible polynomial. What are the strategies for showing that this isn't generated by one element as a $k$-algebra?
Matt
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Image of the map induced on spectra

Apologize in advance if this is a bit trivial but I am stuck on the following: Prove that for $\varphi : R \to S$ a map between commutative rings, the prime $\mathfrak{p}$ is in the image of the induced map $\mathrm{Spec}(S) \to \mathrm{Spec}(R)$…
Filip Chindea
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Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$

Suppose $R$ is integral domain and $K$ is the fraction field of $R$. Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$.
m a s
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How to "Hom" the canonical sequence in a ring

I'm stuck with the dual of an exercise in Atiyah-MacDonald. It's easily seen that tensoring over $R$ with $-\otimes M$ the sequence $$ 0\to \mathfrak a\to R\to R/\mathfrak a\to 0 $$ one gets the isomorphism $R/\mathfrak a\otimes M\cong…
fosco
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proof of Proposition 3.3.18 in Bruns and Herzog

This set of questions pertains to the proof of Proposition 3.3.18(b) in Bruns and Herzog, Cohen-Macaulay Rings: Question 1: It seems to me that under the hypothesis (a) of the theorem, the torsion-freeness of $\omega_R$ forces the underlying ring…
Manos
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Krull dimension and zero divisors of $k[x,y,z]/(x^ay,x^bz)$

I found the primary decomposition of $(0)$ in the ring $k[x,y,z]/(x^ay,x^bz)$, where $a\geq b \geq 1$, $k$ is alg. closed, to be $(x^b) \cap (x^a,z) \cap (y,z)$ (is this correct?). Now I am now looking for the zero divisors of the ring, and I…
baltazar
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Presentation of a local complete intersection

What is the simplest example of a local (noetherian) complete intersection ring $R$ that can not be presented as $R=S/I$, where $S$ is a regular local ring and $I$ is an ideal generated by a regular sequence? Note that any definition of a local…
Alex
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Given an ideal of a ring $R$, is there any way by which the associated primes of $R/I$ can be computed without knowing a primary decomposition of $I$?

Suppose I've been given an ideal $I$ of a commutative ring $R$ and I don't know the primary decomposition of $I$. How do I find the associated primes of $R/I$? Please give some approach if possible. More specifically, I've been trying to find…
adrija
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Question about modules of finite length

This must be very easy, but I'm somehow unable to see it: Let $M$ be a module of finite length over a noetherian local ring $A$ with residue field $k$. If $M$ is nonzero, then there exists a surjective $A-$module homomorphism $M \rightarrow k$. How…
Cyril
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Inverse image of a maximal ideal under a morphism of finitely generated $\mathbb{C}$-algebras.

Let $$ f: A\to B $$ be a morphism of finitely generated $\mathbb{C}$-algebras, suppose $\mathfrak{m}\unlhd B$ is a maximal ideal, I want to show that $f^{-1}(\mathfrak{m})$ is a maximal ideal of $A$. Consider the morphism $$ A\to B/\mathfrak{m},…
Jimmy R
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