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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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A question on localization

For an integral domain $R$, I know that $$R=\bigcap_{\text{maximal ideals }\mathcal{m}}R_{\mathcal{m}}.$$ Why must $R$ be an integral domain? I want to know a counterexample when $R$ is not an integral domain.
Sang Cheol Lee
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Is a prime ideal lying between associated primes also an associated prime?

Let $A$ be a ring and consider three prime ideals in $A$ with inclusions $P\subset Q\subset P'$. Suppose that $P,P'\in \operatorname{Ass}(A)$. Must then also $Q\in \operatorname{Ass}(A)$?
evgeniamerkulova
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Quotient of Dedekind rings by a power of a prime ideal

Let $ A $ be a Dedekind ring, let $ P$ be a nonzero prime ideal of $ A $. What can we say about $ P^n $ or $ A/P^n $ in general? For example, is it true that $ A/P^n $ is a vector space over $ A/P $ of dimension $ n $?
user228960
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Homomorphic image of ring

I read Craig Huneke's paper "Hyman Bass and Ubiquity: Gorenstein Rings", in which he gave a definition. "Let $S$ be a polynomial ring and $R$ be a homomorphic image of $S$ of dimension $d$." Then does he mean that $R$ is just a subring of $S$. I…
T C
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proof: if $K(x_1,\ldots,x_k)/K$ is a field extension, and the $x_j$ are algebraically independent over K, M cannot be a finitely generated K-algebra

I'm trying to prove the following (as a part of the proof of Zariski's Lemma): Let $K$ be a field. $M = K(x_1,\ldots,x_k)$, $x_1,\ldots,x_k$ algebraically independent over K. Then $M$ cannot be a finitely generated $K$-algebra. Proof: Assume for a…
mike
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To show that $\mathfrak{m}B \neq B$ for all maximal ideals $\mathfrak{m} \subset A$, where $B$ is finitely generated as $A$-module

Let $B \supset A$ be an extension of commutative rings such that $B$ is finitely generated as $A$-module. Prove that $\mathfrak{m}B \neq B$ for all maximal ideals $\mathfrak{m} \subset A$. I have no idea how to proceed. I think it should be enough…
Naweed G. Seldon
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Real rings with a zero divisor

Let $A$ be a commutative ring with identity. We say that $A$ is a real ring if every sum of squares of non-zero elements of $A$ is not zero. It is well-known that if $A$ is a real ring and an integral domain, then the quotient field of $A$ is an…
Learner
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Orders of vanishing and valuation

In https://stacks.math.columbia.edu/tag/02MB，we have following definition： Definition 10.120.2. Suppose that $K$ is a field, and $R \subset K$ is a local Noetherian subring of dimension $1$ with fraction field $K$. In this case we define its order…
user395911
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Question on dimension

In some text book of Commutative Algebra, the authors defined the height of an ideal $I$ of a commutative ring $R$ is the maximal of length of a prime ideal chains : $\mathfrak{p}_{0}\subset \mathfrak{p}_{1}\subset...\subset…
morse
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Is the localization of such a Noetherian domain at every maximal ideal Artinian?

My question here is inspired by this question here. I am not asking how to prove the problem there, but if an alternative approach is possible. Suppose $A$ is a Noetherian integral domain in which for every maximal ideal $\mathfrak{m}$, the…
user38268
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Why are integral elements defined with monic polynomials?

Why are integral elements defined in terms of monic polynomials? Why do we wish to split the cases between non monic polynomials and monic polynomials? I.e, algebraic elements and integral elements. Motivation?
green frog
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Best approach to comparing various paths in commutative diagrams: equality or equivalence

I'm an engineering graduate student who is involved in control engineering research. I've just recently started to extract some (relatively) abstract properties from the theory that I am developing, and I'm not sure how to compare different…
user240612
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Show that canonical $R$-linear map $\tau:M\otimes_R N\to M\otimes_{R'} N$ is isomorphism if $R'$ is a quotient of $R$

I am having a problem with exercise 8.9 in A Term of Commutative Algebra by Altman and Kleiman. The exercise goes as follows: Let $R$ be a ring, $R′$ an $R$-algebra, $M$, $N$ two $R′$-modules. Show there is a canonical $R$-linear map $\tau :…
Student G
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Non-flatness of the map $\mathbb C[s,t]\to \mathbb C[x, y]$ sending $s$ to $x$ and $t$ to $xy$.

I would like to ask if there is a simple way to prove the non-flatness of the above morphism of rings using just the definition of a flat module.
agleaner
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Idea about syzygy

I am studying commutative algebra by myself. I got stuck to understand some concepts like syzygy. I am looking for a good reference for that even if it is a paper. Any help will be appreciated.
Team
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