# Tagged Questions

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

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### Fraction field of the formal power series ring in finitely many variables

What is the fraction field of the formal power series ring over a field in finitely many variables $K[[X_1,\dots,X_n]]$? Is there a nice description for this field? When $n=1$, I know this is the ...
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### Localisation contained in completion?

I'm working on an exercise in which I have to show that localising and completing are exact functors. More precisely I have a Dedekind domain $R$ and a prime ideal $\mathfrak{p}$ and I have to show ...
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### On unique factorizations of ideals

Using standard notations, let $K$ be a number ﬁeld and $S = \left\{p_{1}, ..., p_{n}\right\}$ a ﬁnite set of non-zero prime ideals of $K$. Let $a$ be a non-zero fractional ideal of $K$. Prove that ...
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### Different version of Gauss's Lemma

Let $A$ be a domain with field of fractions $K$. Let $f, g \in A[X]$ with $g$ monic. Show that if $f/g \in K[X]$ then $f/g \in A[X]$. So I tried the direct approach by just assuming $f/g$ has a ...
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### Projective Spectrum of $K[X,Y]$

Let's assume that $K$ is algebraically closed. I'm having some difficulties figuring out what $\text{proj}\;K[X,Y]$ is, where $K[X,Y]$ is interpreted as a graded ring. Any hints? So far I have only ...
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### Ideal generated by a irreducible element

Is the ideal generated by an irreducible element always a prime ideal in a ring? If so why?
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### If $A$ is a Dedekind domain and $I \subset A$ a non-zero ideal, then every ideal of $A/I$ is principal.

In this question I will use the following definition of a Dedekind domain: An integral domain $A$ is a Dedekind Domain if: 1) $A$ is a Noetherian Ring. 2) $A$ is integrally closed. 3) Every non-...
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### epimorphism in the category of commutative rings

Let $\phi:A\to B$ be an epimorphism in the category of commutative rings, we can find that the induced continuous map $\phi^*$ from Spec$B$ to Spec$A$ is injective as a map between sets, I want ...
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### Particular case of Nakayama's lemma

I'm trying to prove the following particular case of Nakayama's lemma. Let $R$ be a commutative ring and $a\in R$ be nilpotent (let's suppose $a^{k-1}\not=0$, $a^k=0$). Then $aM=M \Rightarrow M=\{0\}$....
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### Motivation behind the definition of Prime Ideal

Can someone explain what's the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?
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### What is a linear resolution?

Can anyone tell me where I may find an introduction to linear resolutions (of a $k[x_1,\ldots,x_n]$-module or ideal) including, of course, the standard definition of such a resolution, as well as its ...
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### *writing* proofs involving commutative diagrams

This question is a little fuzzy so might be closed, but I'll give it a shot. I'm sorry this question has quite a long introduction, I don't see how to formulate it more concisely. In modern algebraic ...
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### Example of non-Krull integrally closed BFD?

Here's another question in the same spirit as my previous one: Are there any integrally closed BFDs which are not Krull domains? Some background information: A BFD (bounded factorization domain) is ...
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### Example of non-Noetherian non-UFD Krull domain?

After a confusing session of hopping through Wikipedia articles, I started trying to summarize for myself some of the inclusions and relations among the many types of integral domains. Right now I'm ...
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### Involutions on commutative rings

I found that all the commutative rings with involution I know are the following: complex number with complex conjugation (plus similar constructions based on rationals and its extensions), any ...
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### Going-up property

Given commutative rings $A$ and $B$, if $B$ is an $A$-algebra, under what conditions, other than $B$ being integral over $A$, will the Going-Up property hold? Is there a condition weaker than ...
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### Explicit examples of infinitely many irreducible polynomials in k[x]

My question is the following. Is it possible to give examples of infinitely many irreducible polynomials in a polynomial ring $k[x]$ with $k$ a field? I'm interested in this because I'm ...
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### The ring of germs of functions $C^\infty (M)$

Define $C^\infty (M)_x := \{ (U,f) | x \in U$ open $, f \in C^\infty (U) \} / \sim$ where $M$ is a manifold and $(U,f) \sim (V,g)$ if $\exists W$ open, $x \in W$ such that $W \subset V \cap U$ ...
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### Determining the minimal number of generators of the maximal ideal of a local Noetherian ring

Let ($A,m$) be a local, Noetherian ring. If $n$ is the minimal number of generators of the unique maximal ideal $m$, then by Krull's Hauptidealsatz and Nakayama's Lemma, we have the following ...
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### When is an affine algebra normal?

I was looking at results that give necessary (and possibly sufficient) conditions on an ideal $I$ in the ring $R=k[x_1,x_2,...,x_n]$ such that $R/I$ is normal. I don't know if there is a standard ...
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### If $Rm$ is free and $N$ is free is $m \otimes n \neq 0$?

This post is a follow up to the counterexample presented in the following questions If $Rm$ is free, how do you show $m \otimes n \neq 0$?. The hope now is that we can eliminate the pathological ...
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### Showing $R(m\otimes n)$ is free if $Rm$ and $Rn$ are free

Let $R$ be a commutative ring with identity. The following is a statement I came across about the submodule $Rt$ generated by a decomposable tensor $t=m\otimes n$ being free, given that $Rm$ and $Rn$...
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### Generators for $M_n(\mathbb Q)$

What is the minimum number of generators for $M_n(\mathbb Q)$, the set of $n \times n$ matrices over $\mathbb Q$, which will generate it as an algebra over $\mathbb Q$ ?
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### Injective map between power series ring

Suppose $k$ is a field and let $n > m$. Does there exist injective homomorphisms $$k [[x_1, x_2, \ldots, x_n]] \rightarrow k[[x_1, x_2, \ldots, x_m]]\ ?$$
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### Does maximal Cohen-Macaulay modules localize?

Let $A$ be a Noetherian local ring and $M$ a finitely generated $A$-module such that $$\operatorname{depth}M= \dim M=\dim A.$$ I can prove that \operatorname{depth}M_{\mathfrak{p}}= \dim M_{\...
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### The valuation ring $R$ in $K(T)$, such that $K[T] \subsetneq R \subsetneq K(T)$

$K$ is an algebraically closed field, $K[T]$ is the ring of polynomials of one indeterminate over $K$, and $K(T)$ is its field of fractions. A valuation ring $R$ in $K(T)$ which includes $k[T]$ and is ...
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### Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
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### A wrong proof about Dedekind domains

I "proved" that a Dedekind domain is a PID, but as we know this is wrong (for example $\mathbb{Z}[\sqrt{-5}]$). I do not know what is wrong in my proof: Suppose $R$ is a Dedekind domain, $I$ is ...
Let A be a Noetherian ring, Is the set of prime ideals $\{p\in \operatorname{Spec} A| A_p$ is a reduced ring $\}$ an open subset of $\operatorname{Spec} A$ in Zariski topology?
For an affine variety $X=V(x^{2}+y^{2}-1, x-1)$, I found the ideal of $X$, $I(X)=\langle x-1,y\rangle$. But I don't know $I(X)=\langle x^{2}+y^{2}-1, x-1\rangle$.