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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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If ${A}$ is a finitely generated $k$-domain, why is the field of fractions of ${Frac(A)}$ a finitely generated field extension of $k$?

In theorem ${3.2}$ of Harsthorne, part $d$ is implicitly using the fact that if ${A}$ is a finitely generated $k$ domain then ${Frac(A)}$ is a finitely generated field extension of $k$. Why is this the case?
CoffeeBean
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If $B = \widehat{A[x]}$ with $x \in B$ and $A = \widehat A$, does it follow that $B = A + xB$?

At the beginning of Proposition 3.4.4 in Caruso's An introduction to $p$-adic period rings, it is simply stated that because $A_{\mu_0}$ is the $p$-adic completion of $A_{\inf}[\frac t p]$, it follows that $A_{\mu_0} \subset A_{\inf} + \frac t p…
mesz
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Internal Direct Sum of Finitely Generated Submodules is Finitely Generated

Claim: Suppose $M$ is the internal direct sum of $M_1 , \ldots , M_n \;$. $M$ is a finitely generated $R$-module iff $M_1 , \ldots , M_n $ are finitely generated $R$-modules. Here $M$ is a $R$-module, where $R$ is commutative, and $M_1, \ldots ,…
El Spiffy
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Implications from the proof of Atiyah Macdonald Proposition 8.8

Proposition 8.8 from Atiyah Macdonald states: Let $A$ be an Artin local ring. Then the following are equivalent: every ideal in $A$ is principal; the maximal ideal $\mathfrak{m}$ of $A$ is principal; $\dim_k(\mathfrak{m} / \mathfrak{m}^2) \leq…
Zanjo
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Homomorphism extension to fields based on integral extension

Let $R$ be a domain, $R\subset S$ is an integral extension of rings. Suppose $0\neq s\in S$ such that for any $r\in R$ and any integer $n$, $rs^n=0$ implies $r=0$.Then there exists $0\neq x\in R$, such that for every ring homomorphism $\phi : R\to…
Display name
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A finitely presentable real algebra without morphisms to the real field and without morphism from complex field?

I guess there must standard examples but I can't seem to find them. I tried ${A = \mathbb{R}[x,y]/(x^2 -1, y^2 x + x)}$, which is not the domain of a homomorphism to $\mathbb{R}$ because it would have to send $x$ to $1$, and so $y^2$ to $i$. But…
Boogie
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Quotients of Integral Extensions are Integral.

If $A \subset B$ is an integral extension, then $\frac{A}{I} $ is integral over $\frac{B}{J}$ where $I=J \cap A$. I was able to prove this but I am wondering why can't be generalise it for any ideal of $A$? I was looking for some counterexamples in…
permutation_matrix
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why is $\bar u \circ \bar v =v \circ u \circ f?$

The proposition 2.9 of Atiyah and Macdonald : It is written that The sequence $$M'\xrightarrow u M \xrightarrow v M'' \rightarrow 0$$ is exact iff for all $A$-modules N, the sequence $$0\rightarrow Hom (M'',N)\xrightarrow{\bar{v}}…
wasiu
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Integrally Closed but not UFD

We know that UFDs are integrally closed. But there is no theorem which states that Integrally Closed are UFD. So I wonder there should be a counter-example. I know that $Z[\sqrt{4k+2}$ and $\mathbb{Z}[\sqrt{4k+3}$ are integrally closed. I wonder…
permutation_matrix
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Is $R[X_1,X_2,\cdots]/\langle X_1,X_2,\cdots\rangle$ finitely presented?

This is exercise 5.29 from A Term of Commutative Algebra by A. Altman & S. Kleiman. A digital version may be found here. The statement of the exercise is the following. Let $R$ be a ring, $X_1,X_2,\cdots$ infinitely many variables. Set…
Ze Chen
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A kind of ideals in Gorenstein local rings.

Let $(R,m)$ be a Gorenstein local ring. Then there exists an irreducible ideal $I\subset R$, such that $\sqrt I=m$, and $\operatorname{pd}_RI<\infty$. I am looking for such $I$. Clearly, if such $I$ exists, $R/I$ has to be Gorenstein. And as an…
Bromelain
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$0 \rightarrow{}2\mathbb{Z} \xrightarrow{f} \mathbb{Z} \xrightarrow{p} \mathbb{Z}/\mathbb{2Z} \to 0$ is an exact sequence .True/False

Is the following statement True/False The sequence $0 \rightarrow{}2\mathbb{Z} \xrightarrow{f} \mathbb{Z} \xrightarrow{p} \mathbb{Z}/\mathbb{2Z} \to 0$ is an exact sequence where $f$ is the inclusion map and $p$ is the projection of …
wasiu
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Linear algebra over UFDs

In linear algebra over a field $K$ as set of column vectors $v_1,\dots,v_k\in K^n$ is a basis of the subspace $V$ iff $v_1,\dots,v_k\in V$, and the matrix $[v_1,\dots,v_k]$is of rank $k$. I wonder under what circumstances something similar can be…
HCH
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Why is $M$ faithful as an $A/{\alpha}$ module?

I have some confusion on Atiyah Commutative algebra On Page No $20$ it is written that An $A-$Module is faithful if $Ann(M)=0$.If $Ann(M)=\alpha$ ,then $M$ is faithful as an $A/{\alpha}$ module My confusion : Why is $M$ faithful as an…
wasiu
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Left exact contravariant morphism is an isomorphism for fin. pre. modules

I was reading the proof of the following proposition: Proposition. Let $G$ and $H$ be covariant or contravariant functors from $A$-mod to $B$-mod, and let $\theta: G \rightarrow H$ be a morphism of functors. If $\theta(A)$ : $G(A) \rightarrow H(A)$…
PCeltide
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