Questions tagged [integral-extensions]

This tag is for questions regarding to the Integral Extensions or, Integral Ring Extensions.

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integral extension and existence of finitely generated faithful module

I am reading the book Advanced modern algebra by Rotman. In the integral extension section, he claims that if $R^*/R$ is a ring extension and $u\in R^*$ be a nonzero element then If there is a ...
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50 views

Galois group of $x^5-x-1$ over $\Bbb Q$ using integral extension ring theory

Let $f= x^5 − x − 1$ and $L$ be the splitting field of $f(x)$ over $\Bbb Q$. Suppose $B$ is an integral closure of $\Bbb Z$ in $L$ and $P$ is a maximal ideal of $B$ such that $P \cap \Bbb Z = 2\Bbb Z$...
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If a finite group acts on an integral domain which is integrally closed, then the fixed point subring is also integrally closed

Let $G$ be a finite group acting on an integral domain $R$, and let $S$ denote the fixed point subring: $$ S=\{x\in R: gx=x \text{ for all }g\in G\}$$ I am asked to show that $R$ is integral over $S$, ...
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25 views

On the “largest ring extension” of a Noetherian domain inside fraction field that is module finite

Let $R$ be a Noetherian domain with fraction field $K\ne R$. Let $\overline R$ be the integral closure of $R$ in $K$. Let us call that a subring of the fraction field $R\subseteq R^{fn}\subseteq K$ ...
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Is the polynomial ring of elementary symmetric polynomials involving n variables over a field is integrally closed?

Let $\mathbb{F}$ be any field and $x_1,\ldots,x_n$ be algebraically independent over $\mathbb{F}$ and also let $s_i=i$th elementary symmetric polynomial in the $x_i's$, e.g., $s_1=x_1 + \cdots + x_n;\...
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34 views

On the colon ideal of a torsion-free module inside it's reflexive hull

Let $(R,\mathfrak m)$ be a Noetherian local complete domain of dimension $1$, with fraction field $K$. (all these assumptions on $R$ imply in particular that for every finitely generated $R$-algebra ...
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56 views

On integral closedness of the multiplication of a monomial integrally closed ideal with the homogeneous maximal ideal

Consider the monomial ideal $I=(x^d, y^az^{d-a})$ in $\mathbb C [x,y,z]$ where $1\le a\le d-1$ are integers. Let $\mathfrak m=(x,y,z)$. Let $J=\overline I$ be the integral closure of $I$. If $J\...
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50 views

On the intersection of integral closure of powers of an ideal

Let $R$ be a commutative Noetherian ring. Let $J$ be a proper ideal of $R$ . Let $Min(R)$ be the set of minimal primes of $R$ (this set is finite as $R$ is Noetherian). Then how to prove that $\cap_{...
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39 views

The stalk of the image sheaf on a normalization curve

My question is based on an exercise (Ex II.6.9) in Hartshorne's GTM52. Suppose that $X$ is a projective curve over $k$ (an algebraically closed field). Let $\tilde{X}$ be its normalization and $\pi: ...
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73 views

In prime characteristic, is being $N$-1 a local property?

A Noetherian domain $R$ with fraction field $K$ is said to satisfy $N$-1 if the integral closure of $R$ in $K$ is module finite over $R$. My question is: Let $R$ be a Noetherian domain of finite ...
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26 views

If every subring of $S$ containing $v\in S$ is ring-finite over $R,$ is $v$ necessarily integral over $R$?

In the following, $R$ is a subring of a ring $S,$ and $v\in S.$ All of my rings are commutative with $1.$ I like to use the following terminology. Definitions. I say that $S$ is module-finite over $...
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58 views

Is the normal fiber cone of any $\mathfrak m$-primary ideal Noetherian?

For an ideal $I$ in a commutative Noetherian ring $R$, let $\overline I$ be the integral closure of $I$. Now for an ideal $I$ in a Noetherian local ring $(R, \mathfrak m)$ consider the graded ring $\...
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31 views

Polynomial ring over an analytically unramified ring is locally analytically unramified

Let us call a Noetherian local ring $(R, \mathfrak m)$ to be analytically unramified if the $\mathfrak m$-adic completion of $R$ is reduced i.e. has no non-zero nilpotent. https://en.m.wikipedia.org/...
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50 views

Extending “the ring of exponents” from $\Bbb{Z}$ to that which contains a solution to $4^x - 3 = 0 \pmod 5$?

Let $R = \Bbb{Z}/5\Bbb{Z}$ be the ring of integers modulo $5$. Then $4^x = 1, 4, 1, 4, 1, 4, \dots$ as $x$ ranges over $\Bbb{Z}$. Thus $4^x = 3$ and $4^x = 2$ have no solution $x \in \Bbb{Z}$. ...
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61 views

Is $R[X,Y]/(XY-1)$ a finitely generated $R[X]$-module?

Let $R$ be a commutative Noetherian ring. Consider the localisation of $R[X]$ at the multiplicative closed set $\{1,X,...\}$ i.e. $R[X]_X \cong R[X,Y]/(XY - 1)$. Notice that the localisation map $R[X] ...
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32 views

Going Up Theorem

Let $R^*/R$ be a ring extension with $R^*$ integral over $R$. If $p\subset q$ are prime ideals in $R$, and if $p^*$ is a prime ideal in $R^*$ lying over $p$, then there exists a prime ideal $q^*$ ...
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107 views

Computing Hilbert-Samuel multiplicity of $k[[f_1(t),…,f_n(t)]]$

Let $k$ be a field and $k[[t]]$ be the formal power series ring over $k$. Consider the subring $R=k[[f_1(t),...,f_n(t)]]$ for some $f_1(t),...,f_n(t)\in k[[t]]$ such that $k[[t]]\subseteq Q(R)$ and ...
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37 views

A question about integral ring extension

Lemma: Let $R\subset S$ be an integral ring extension. Let $I$ be a proper ideal of $R$. Then $IS$ is a proper ideal of $S$. I'm reading the proof from following link http://www.math.rwth-aachen.de/~...
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88 views

On the intersection of integral closures of all powers of maximal ideal

Let $(R,\mathfrak m)$ be a Noetherian local ring of positive dimension. For an ideal $J$ of $R$, let $\bar J$ denote the integral closure of $J$ (https://en.m.wikipedia.org/wiki/...
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98 views

Easy examples of integrally closed non-factorial domains?

I'm looking for some easy to prove examples of Noetherian domains which are not UFD but are integrally closed in their fraction field. The common examples I know (like $\mathbb Q[x,y]/(x^2+y^2-1)$ ...
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37 views

Lying over property in Nullstellensatz theorem

In one of our notes we proved the Nullstellensatz theorem (fg algebra $L$ over a field $k$ which is also a field is a finite extension) by Noether-normalizing and assuming the middle ring B (the one ...
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53 views

Concerning the ring extension $A[a^{-1}] \subseteq B[a^{-1}] \subseteq C[a^{-1}]$

Let $k$ be a field of characteristic zero and let $A \subseteq B \subseteq C$ be three integral domains (= commutative rings without zero divisors) which are $k$-algebras. Denote the field of ...
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71 views

On the integral closure of Noetherian local domain in a finite extension field of the fraction field

Let $(R, \mathfrak m)$ be a Noetherian local domain such that the $\mathfrak m$-adic completion of $R$ is also an integral domain. Let $K$ be the fraction field of $R$ . Let $L$ be a finite extension ...
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1answer
70 views

Integral closure of pull-back of an ideal via a birational, finite morphism of rings

Let $R,S$ be Noetherian normal rings (i.e. they are locally normal domains at every prime ideal, so in particular they are reduced). Let $f: R \to S$ be a ring homomorphism such that via this map, $S$ ...
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75 views

When is the trace of an integral element of an algebra integral?

Let $R$ be an integral domain, $K$ its field of fractions and $A$ a (unital, associative) $K$-algebra of finite dimension. Call $a \in A$ integral (over $R$) if it is a root of a monic polynomial with ...
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51 views

Existence of maximal ideal lying over and residue of element equals root of minimal polynomial

Suppose $A \hookrightarrow B$ is a finite injective Ringhomomorphism, where $A$ is a UFD and $B$ is an integral domain. Let $f \in B$ and $P=T^n + a_{n-1} T^{n-1} + \cdots + a_0 \in A[T]$ be ...
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99 views

Is $\mathbb{Z}[x,y]/(x^3+px+q - y^2)$ integral over $\mathbb{Z}[x]$?

Let $ B = \mathbb{Z}[x,y]/(x^3+px+q - y^2)$ and $A = \mathbb{Z}[x]$. I want to know whether the ring extension $A \subset B$ is integral or not. There can be two possibilities: $x^3+px+q$ is ...
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144 views

$\mathbb{R}[x,y]/(x^2+y^2-1)$ is integrally closed.

I have been thinking about how to prove that the integral domain $\mathbb{R}[x,y]/(x^2+y^2-1)$ is integrally closed and I am writing this post to ask for proof verification. Attempt. Let $R=\mathbb{...
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45 views

Integral extension of domains implies algebraic extension of their fraction fields

If A $\subseteq B$ are domains and $B$ is integral over $A$, then is $Frac(B)$ algebraic over $Frac(A)$? Ia it because of the following: Any element of $Frac(B)$ is of the form $b/t$ with $b,t \in B,...
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123 views

Prime ideals lying over a P.I.D

Let $R$ be a P.I.D, and let $q$ be irreducible in $R$. If we have the extension $R'$ given by: $R'= \Big\{ a+\eta \cdot b: \; a,b\in R\}$ such that $\eta\in R'$ satisfies $\eta^2=q$. I'm trying to ...
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Need help in understanding proof of a statement regarding 'integral closure' of a Noetherian integrally closed domain

In Page No. $13$ of Serre's Local Fields, we have the following paragraph: $K$ is a field, and $L$ a finite extension of $K$. $A$ is a Noetherian integrally closed domain, and $K$ is its field of ...
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35 views

Why is the map $\varphi$ closed?

Let $S/R$ be an integral extension. Consider the map $\varphi : \text {Spec} (S) \longrightarrow \text {Spec} (R)$ defined by $$\varphi (\mathfrak B) = \mathfrak B \cap R,\ \mathfrak B \in \text {Spec}...
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46 views

How does $(\mathfrak {B_2} / \mathfrak {B_1}) \cap (R/ \mathfrak p) = (0)$?

Let $S/R$ be an integral extension. Let $\mathfrak {B_1}, \mathfrak {B_2} \in \mathrm {Spec} (S)$ with $\mathfrak {B_1} \subset \mathfrak {B_2}$ are given. If $\mathfrak {B_1} \cap R = \mathfrak {B_2} ...
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25 views

An equation defined by norm

Let $f$ be an Eisenstein polynomial of degree $n$ and the prime $p$. $\alpha$ is a root of $f$. Let $\mathbb{Q}(\alpha)=K$, Prove that for any $\gamma\in O_K$, there exist $a\in \mathbb{Z}$, such that ...
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35 views

Example of integral extension of degree $p$?

For a prime $p$, does there exist an example of a (pair of) domains (if yes any way to construct?) $R$ and $S$ such that Both $R$ and $S$ are normal complete local domains $S$ contains an alg. ...
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46 views

$K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension

Let $K$ be a field. Prove that $K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension. I know that $K[x,y]/(xy-1) \simeq K[t, t^{-1}]$, but I'm not sure if this would be useful to prove the ...
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128 views

If $Q \in\mathrm{Spec}(B)$ is the unique prime ideal lying over $P \in\mathrm{Spec}(A)$, then $B_P=B_Q$.

Let $A \subseteq B$ be an integral extension of integral domains. Suppose $P \in \mathrm{Spec}(A)$ and $Q \in \mathrm{Spec}(B)$ is the unique prime ideal lying above $P$. Prove that $B_Q=B_P$. My ...
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73 views

For an integral extension $A \subseteq B$, $\sqrt{IB}\cap A=\sqrt{I}$

Let $A \subseteq B$ be an integral extension and $I \subseteq A$ an ideal. Prove that $\sqrt{IB}\cap A=\sqrt{I}$. I saw a similar property holds for Jacobson radical where the following two facts ...
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54 views

Integral extension of local rings of projective varieties

Let $X$ and $Y$ be projective varieties and $φ\colon X \to Y$ a regular birational map. If $q\in Y$ and $φ^{-1}(q)$ is finite and $φ(p) = q$, is it true that $\mathcal O_{X,p}$ is an integral ...
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29 views

$A \subset B$ be integral and flat extension then it is faithfully flat.

I want to prove the following: $A \subset B$ be integral and flat extension of rings then it is faithfully flat. Clearly enough to show that for every ideal $I$ of $A$, $I^{ec}=I$. Since the ...
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57 views

Associated prime ideals of integral extension

Let $R$ be a commutative ring with unit element and $S$ be an integral extension of $R$. Then when will $\mathrm{Ass}_R(R) = \mathrm{Ass}_R(S)$? I am not able to prove $\mathrm{Ass}_R(S) \subseteq \...
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158 views

$K[X]_X$ is not integral over $K[X]$

The localization $K[X]_{X}$ is a ring extension of $K[X].$ I want to show that $K[X]_X$ is not integral over $K[X]$ using lying above. I tried to find a maximal ideal in $K[X]_X$ whose contraction in ...
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132 views

Contraction of a maximal ideal is maximal in a particular case.

I am reading through Stephen McAdam's Asymptotic Prime Divisors. I am stuck on Lemma 3.1, which states: Lemma 3.1. Let $R$ be a Noetherian domain with integral closure $\bar{R}$. Let $(V,N)$ be a D. ...
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45 views

On Noetherian domains whose proper submodules of the fraction field are projective

Let $R$ be a Noetherian domain with fraction field $K$. If every proper $R$-submodule of $K$, containing $R$ (i.e. all $R$-submodules $M$ of $K$ such that $R \subseteq M \subsetneq K$) is projective ...
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186 views

The set of integral elements form a ring.

Let $A \subset B$ be two rings. I know that an element $x \in B$ is integral over $A$ iff $A[x]$ is contained in a finitely generated $A$-module $T \subset B$. I also know that if $b_1,...,b_n$ are ...
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36 views

When is $F[X]/(f)$ over $F$ an integral extension?

I was looking at the concept of integral extension. So $K/F$ is integral means that for every $a \in K$ is the root of a monic polynomial with coefficient in $F$. For example $\mathbb{Q}/\mathbb{Z}$ ...
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64 views

On integral closedness of formal power series ring over an integrally closed domain satisfying Krull intersection principle

Let $R$ be a normal domain (i.e. an integral domain integrally closed in its fraction field) such that for every non-unit $t\in R$, $\cap_{n\ge 1} (t^n)=(0)$ ; then is it true that $R[[X]]$ is normal (...
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43 views

If $x$ is algebraic over a quotient field $K$ of $A$, then there exists an integral element $cx$ for some $A \ni c \neq 0$.

Let $A$ be a commutative ring, $K$ its quotient field and $x$ algebraic over $K$. This means that there exists a polynomial $f(X)$ with coefficients in $K$ such that $f(x) = 0$. In other words, if ...
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119 views

Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
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68 views

Geometry of the subalgebra $\Bbbk[x^2-1]\leq \Bbbk[x]$ (intuition for integral elements)

Given a field $\Bbbk$ consider the subalgebra $\Bbbk[x^2-1]\leq \Bbbk[x]$. This is an integral extension of algebras. Write $\mathfrak q= (x-1)\vartriangleleft \Bbbk[x]$ and $\mathfrak p=\Bbbk [x^2-1]\...