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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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Simple minded details about why a particular map is étale

In the category of (commutative, unital) $\mathbb{C}$-algebras, let ${f : \mathbb{C}[t, t^{-1}] \rightarrow \mathbb{C}[t, t^{-1}, x]/(x^2 -t)}$ be the obvious map sending $t$ in the domain to $t$ in the codomain. Let $N$ be a nilpotent ideal in…
Boogie
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Two ideals with equal radical in a noetherian ring

Let $A$ be a commutative noetherian ring with two ideals $I,J$ such that $\sqrt{I}=\sqrt{J}$. Does there always exist integers $p,q,r$ such that $$ I^p \subset J^q \subset I^r? $$
Bonanza
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Reduction of a $\mathbb{Z}$-algebra

Let $R$ be the commutative unital ring $\mathbb{Z}[x, y, z]/(x^3-y^2-1728z)$. Let $p$ be a prime number, assume $p>3$. Is it true that $R\otimes_{\mathbb{Z}} \mathbb{F}_p\approx \mathbb{F}_p[u, v]$? My argument is that when we reduce the relation…
user693243
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minimal primes of a homogeneous ideal are homogeneous

I am trying to study the proof of this result. It appears as part 3 of the proposition on page 2 of the following document http://math.mit.edu/classes/18.721/projgeom6.pdf I understand everything but the last line. Here the author says…
user9504
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Necessity of the Noetherian condition to derive a result about associated prime ideals

Let $A$ be a Noetherian ring and $\mathfrak a$ be an ideal of $A$. Then it is well-known that the associated prime ideas of $\mathfrak a$ are those prime ideals that have the form $(\mathfrak a:x)$ for $x \in A$. I want to know whether Noetherian…
Ehsan M. Kermani
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Moving elements in chains of primes of a Noetherian local ring

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d$. Show for any chain of prime ideals $\mathfrak p_0\subsetneq...\subsetneq\mathfrak p_n$ with $a∈\mathfrak p_n$ that there is a chain of prime ideals $\mathfrak…
George
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Zerodivisors and nilpotents in $A/I$

I'm studying primary decomposition in the case of polynomial rings with coefficients in a field. I have defined associate prime ideals of an ideal I as the radicals of the primary ideals appearing in a decomposition. Now, I find everywhere that when…
NeuWolferl
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Localization at $0$?

Jacob Lurie gave a very simple example for primary decomposition: Let $R=\mathbb{Z}$, let $M=\mathbb{Z}\oplus \mathbb{Z}/p$. Then $0=\mathbb{Z}\cap \mathbb{Z}/p$. Here $\mathbb{Z}$ is $p$-coprimary, $\mathbb{Z}/p$ is $0$-coprimary. Further he…
Bombyx mori
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What is the right categorial definition of localisation of a module

Let $A$ be a ring, $S$ be a multiplicative subset, $M$ an $A$-module. let $\iota : A \to S^{-1}A$ be the map $a \mapsto a/1$. $\iota$ can be defined categorically as an initial object in the category of ring homomorphisms whose domain is $A$ which…
DBr
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A question on local rings

I was trying to get a counterexample of this fact: given a ring $A$, $f\in A$ and $S=\{1,f,f^2,...\}$, is $S^{-1}A$ always a local ring? Could you help me please? Thank you.
Corra
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Completion of a polynomial ring w.r.t. a maximal ideal

Let $R=k[X_1,...,X_n]$ be a polynomial ring over the field $k$ and let $\mathfrak m$ be a maximal ideal of $R$. Question: Is the $\mathfrak m$-adic completion of $R$ a local ring ? If $\mathfrak m=(X_1-a_1,\ldots,X_n-a_n)$ with $a_i \in k$ then…
Ralph
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How can I show some rings are local.

I want to prove $k[x]/(x^2)$ is local. I know it by rather a direct way: $(a+bx)(a-bx)/a^2=1$. But for general case such as $k[x]/(x^n)$, how can I prove it? Also for 2 variables, for example $k[x,y]/(x^2,y^2)$ (or more higher orders?), how can I…
Gobi
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Locally a domain and connected implies a domain

Let $R$ be a commutative ring with unit. Let $R_p$ be a domain for all $p\in SpecR$ and let $SpecR$ be connected. Is it true that $R$ is a domain or can someone provide a counterexample. Note here that $R$ is not necessarily a Noetherian ring. For a…
user7254
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In GCD domain every invertible ideal is principal

This is the exercise in the book Commutative Rings by Kaplansky. Prove that in a GCD domain every invertible ideal is principal. I'm looking for some hints. Edit After understanding the hint, here is my approach: Let $I$ be an invertible ideal…
Anh_Rose 1210
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Atiyah Macdonald Exercise 5.22

This is a problem in Atiyah Macdonald, Commutative Algebra. Problem 5.22 $S$ is a subring of an integral domain $R$. $R$ is a finitely generated $S$ algebra. If the Jacobson radical of $S$ is $0$, then the Jacobson radical of $R$ is $0$. I…
user45765
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