# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### fraction field of polynomial ring that is a finite extension of the base field

Let $k$ be a field. Let $P$ be a prime ideal of $k[x_1, ..., x_n]$. Let $K$ be a field of fractions of $k[x_1, ..., x_n]/P$. Suppose $K$ is a finite extension of $k$. Does it then follow that $P$ is ...
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### “Pushforward” over flat morphisms of functions which are constant on fibers

I believe the following should be true, but I'm not sure where to find the required commutative algebra to prove it: If $\mathrm{Spec}\,A \rightarrow \mathrm{Spec}\,B$ is a flat morphism of algebraic ...
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### An ideal avoidance

It is known that in a commutative ring $R$ an ideal contained in a finite union of prime ideals $P_i , ( i=1,...,n)$ is a subset of one of them (prime avoidance theorem). Now, if $P_i$'s are arbitrary ...
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### Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
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### Regular Ring is Integrally Closed?

Studying some topics in Algebraic Geometry I've bumped into the following question: Let $A$ be a regular ring. Is $A$ integrally closed? Someone said me that with the hypothesis $A$ local ...
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### $I = (x^2, y^2) ⊂ K[x, y]$; $gin\ (I)=?$

an easy Google search give a lot of results about the definition of generic initial ideal. But all definitions I see, are like this one: I can't use this definition to compute gin(I) even in simple ...
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### Localization of a ring that is not an integral domain

Let $A$ be a commutative ring with unity that is not an integral domain and $\mathcal{P}$ be any prime ideal of $A$. Then I know that $A_{\mathcal{P}}$ is not an integral domain using the ...
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### $I$ is a $J$-primary ideal of $R$ iff $I/L$ is a $J/L$-primary ideal of $R/L$

Let $R$ be a commutative unitary ring and $I$, $J$, $L$ be ideals of $R$ with $L$ proper, $L \subseteq I$ and $L \subseteq J$. A homework question asks to prove that if $R$ is noetherian then $I$ is ...
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### Is there a name for these sequences of subsets of a commutative ring resembling the definition of a graded algebra?

(I am experimenting with writing arrows backwards.) Let $R$ denote a commutative ring. Is there a term for those sequences $A : \mathcal{P}(R) \leftarrow \mathbb{N}$ satisfying the following ...
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### The rank of the integral closure as a free module

Let $O$ be a PID, and let $L$ be a finite separable extension of its quotient field $K$ with degree $n$. Prove that the integral closure of $O$ in the field $L$ is a free module of rank $n$. Here ...
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### What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
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### Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
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### What is the Characteristic Polynomial of an element over a field in this case?

Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and ...
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### Completion of an ideal

On page 109 of Atiyah-Macdonald, the authors let $A$ be a Noetherian ring with an ideal $\mathfrak{a}$. We use the notation $\hat A$ for the $\mathfrak{a}$-dic completion of $A$. They say that we have,...
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### For which $n \in \mathbb{N}$ is it the case that every element of $\mathbb{Z}/n\mathbb{Z}$ is strongly associate to an idempotent?

Definition. Call two elements of a commutative ring associates iff each divides the other. Call them strong associates if there exists a unit that can be multiplied by the first to yield the second. (...
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### Every affine variety in $\mathbb A^n$ consisting of finitely many points can be written as the zero locus of $n$ polynomials

I am reading Gathmann's free online notes on Algebraic Geometry. One exercise asks to show that "Every affine variety in $\mathbb A^n$ consisting of finitely many points can be written as the zero ...
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### “Closure” and “neighborhoods” in Spec(A)

While trying to work through the sequence of problems in Atiyah-Macdonald's first chapter regarding the prime spectrum of a ring, I've run across a small point of confusion. Namely: In the point ...
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Let $A$ be a Noetherian ring and $M$ an $A$-module. I want to show that $M=0$ if $M_P = 0$ for each $P \in \text{Ass}(M)$. Here is my attempt at a solution: Assume for a contradiction that $M \not=... 1answer 178 views ### Depth of a module over local ring and vanishing of Ext functor I'm studying depth of$A$-modules, where$A$is a noetherian ring, in Matsumura's Commutative Algebra text and I'm experiencing some trouble understanding the proof of a basic result. I think all of ... 1answer 74 views ### Calculate the radical of ideals Let$k$be an algebraically closed field and consider$A=k[x,y,z]$. I am supposed to calculate$\text{rad}(x,y)= \{ f \in k[x,y,z] : f^n \in (x,y)\text{for some n} \}$,$\text{rad}(x,z)$and$\text{...
Let $(R,m)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module of dimension $d$. The Krull dimension of $M$ is defined to be the Krull dimension of $R/\operatorname{ann}(M)$. ...