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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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A doubt regarding the proof of "the set of nilpotent elements is an ideal".

On pg.5 of "Commutative Algebra" by Atiyah-Macdonald, Proposition 1.7 states that The set $A$ of all nilpotent elements in a commutative ring $R$ is an ideal. Let $x,y\in A$. Clearly, for any $n\in \Bbb{N}$, $x^n\neq 0$ and $y^n\neq 0$. However,…
user67803
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Localisation via a universal property (Exercise 1.3.D in Ravi Vakil's notes)

On pg 34 of Ravi Vakil's Foundations of Algebraic Geometry he says that an $S^{-1}A$-module is the same thing as an $A$-module for which $s \; \times\; \cdot :M\rightarrow M$ is an isomorphism for all $s \in S$. I can show explicitly that this is…
porkramen
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Some statements on Atiyah-Macdonald Book

A is a DVR, Could anyone help me to explain these statements? I have no idea how to get these conclusions.
Peter
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Two questions on Noetherian domain

Let $A$ be a Noetherian domain of dimension $1$, $\mathfrak a$ is a non-zero ideal in $A$, then $\mathfrak a$ has a minimal primary decomposition $\mathfrak a=\bigcap_{i=1}^n \mathfrak q_i$, where each $\mathfrak q_i$ is $\mathfrak p_i$-primary.…
Peter
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The radical of the intersection need not be equal to the intersection of the radicals

Let $R$ be a commutative ring with 1. Give an example in which the radical of infinitely many ideals is not equal to the intersection of the radicals.
user101689
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A question on Projective modules

If M is a finitely generated projective R-module then $M \bigotimes -$ is exact. I need some help to prove this. So please give some hints.
Germain
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Ring automorphism in an algebraically closed field

Let $K$ be an algebraically closed field and let $\mathfrak{m}$ be a maximal ideal of $K[x_{1},..,x_{n}]$. How to show there is a ring automorphism $f$ of $K[x_{1},..,x_{n}]$ such that: $f(\mathfrak{m}) = (x_1, x_{2},..,x_{n})$ here $()$ denotes the…
user6495
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If $(R,m)$ is Noetherian, $P$ a prime ideal s.t. $R/P$ is 1 dim, then if $x\in m-P$, then rad$(x,P)=m$

If $(R,m)$ is Noetherian, $P$ a prime ideal s.t. $R/P$ is 1 dim, then if $x\in R-P$, then rad$(x,P)=m$. I am looking to prove this statement but I am at a loss how to start. It's part of a proof that I am reading, but this statement appears without…
B M
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A question on Modules

Let $M$ be a module over a ring $A$ and let $f_{1},...,f_{n}$ be elements of $A$ generating the unit ideal. Show that $M=0$ iff $M_{f_{i}}=0$ for $i=1,...,n$. I feel that this is closely related to saying that $X=Spec A$ is quasi-compact. That is,…
user 3462
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Noetherian ring and ascending chain of ideals.

Let $A$ be a noetherian ring, $\mathfrak a_{0}$ an ideal in $A$ and $$S=\{ n\in \mathbb{N}\mid \text{there exist ideals }(\mathfrak a_{i})_{i=1,\dots,n}\text{ such that }\mathfrak a_{0}\subset \mathfrak a_{1}\subset \dots \subset \mathfrak…
user53216
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Definition of 'Maximal Length of Presentation'

In exercise 1.2.6 of the french version (seems like it is also the case of the english version) of Algebre Commutative by Bourbaki, a function $\lambda$ was defined to assign each $R$-module the maximal length of its presentation. Here, the…
SalutaFungo
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When is the module of Kähler differentials reflexive?

Let $A$ be an affine $k$-algebra and $\Omega_{A|k}$ the module of Kähler differentials of $A$? Under what conditions on $A$ is $\Omega_{A|k}$ a reflexive $A$-module? Clearly, $A$ being smooth is sufficient since this is equivalent to $\Omega_{A|k}$ …
HCH
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Atiyah-Macdonald proposition 10.15 $\hat{A}\mathfrak{a}$

I don't understand the symbol $\widehat A \mathfrak{a}$. I saw the following post, but I still have the question. Atiyah-Macdonald, Chapter 10, Proposition 10.15 clarifications There is following answer. Suppose $R\rightarrow S$ is a ring…
nat-cat
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localization over different rings

I guess this is a bit of a soft question. Suppose you have a hom of commutative rings with identity $R\to S$, and a hom of $S$ modules $f: M\to N$. Somehow it shouldn't matter whether you localize $f$ in the category of $S$ modules or in the…
famous mortimer
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An inequality of dimension and depth of Noetherian local rings.

Here is my question: Let $(R,\mathfrak m)$ be a Noetherian local ring with ideal $I\subset\mathfrak m$. Do we have $$\mathrm{depth}_IR+\dim R/I\geq\mathrm{depth}R?$$ If we have, then this is stronger than $\mathrm{depth}R\leq\dim R$ since we have…
WakeUp-X.Liu
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