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Questions tagged [integral-extensions]

https://en.wikipedia.org/wiki/Integral_element#Integral_extensions

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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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1answer
24 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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66 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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1answer
40 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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48 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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1answer
14 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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27 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 $Ass_R(R) = Ass_R(S)$? I am not able to prove $Ass_R(S) \subseteq Ass_R(R)$. Is there a counter ...
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2answers
53 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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1answer
41 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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1answer
38 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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1answer
59 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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28 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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27 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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40 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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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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1answer
57 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]\...
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2answers
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Conceptual reason for some formal properties of integral morphisms of rings

In chapter 5 of Atiyah-Macdonald there are exercises asserting that integral morphisms of rings / integral algebra are stable under tensoring (ex 3) and closed under products (ex 6). The only ...
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1answer
56 views

Noetherian domain whose fraction field is such that some specific proper submodules are projective

Let $R$ be a Noetherian domain (which is not a field) with fraction field $K$. Suppose every proper $R$-submodule of $K$ of the form $R[1/a]$, where $a\in R$, is projective as an $R$-module. Then, I ...
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Showing that $A[x] \cap A[x^{-1}] $ is an integral extension of $A$ when $x$ is invertible and $x$ belongs to a bigger ring containing $A$ [duplicate]

I have a problem- $A, B$ are rings such that $A \subset B$ and $x$ is an invertible element in $B$. I have to show that that the ring $T = A[x] \cap A[x^{-1}]$ is an integral extension of $A$...
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1answer
71 views

Contraction of (non-prime) ideals in integral extensions

If $A \subset B$ is an integral extension, then any prime $p \subset A$ is the contraction of some prime of $B$ (by lying-over property). Does this hold for more general ideals? That is, given an ...
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1answer
53 views

Is an integer extension of a ring the integral closure of this ring in some extension?

This is a question that I just came up with, so it may be completely stupid and I sincerely apologize if that's the case. The question is; Take $A$ to be an integral domain. Let $B$ be an integral ...
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1answer
44 views

If sum and product of elements are integral, they must be integral

I'm asking about a sort of converse to the classical theorem that the integral elements for a ring. Let $A \subset B$ with $A$ a subring of $B$. We know that the elements of $B$ which are integral ...
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1answer
54 views

$A$ is a domain, $Q$ a prime ideal of $A$; connection between integral closedness of $A_Q$ and $A[X]_{Q[X]}$

Let $A$ be an integral domain and $R=A[X]$. Let $Q$ be a prime ideal of $A$ and let $P=Q[X]$. If $R_P$ is integrally closed (in its own fractions field), then is $A_Q$ integrally closed (in its own ...
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1answer
52 views

About the definition of an integral element in commutative rings

Let $R \subset S$ be commutative rings with unity. The usual definition for an element $x \in S$ to be integral over $R$ is to require the existence of a monic polynomial with coefficients in $R$ such ...
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42 views

On contraction of maximal ideals in simple integral extension

Let $R\subseteq T$ be integral domains. Let $a\in T$ be a root of some degree $d$ monic polynomial in $R[X]$. Then of-course $R[a]$ integral over $R$. I am trying to show, without using any heavy ...
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78 views

On the subring $k + Y k [X,Y]$ of the polynomial ring $k[X,Y]$

Let $k$ be a field. Let $B = k[X,Y]$ and $A=k + YB$. So we have the ring extension $ A \subseteq B$. Now I can show that the conductor ideal of $A$ in $B$ is $YB$. So if $Q(A), Q(B)$ are the fraction ...
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1answer
32 views

Sharp's Exercise 13.35

Exercise: Let $R$ be a subring of the commutative ring $S$ and suppose that S is integral over R. i) Show that, if $r$ $\in$ $R$ is a unit in $S$ then $r$ is a unit in $R$. ii) Show that, $Jac(R)=...
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1answer
112 views

How can I compute the discriminant of the field $\mathbb{Q}(\sqrt[3]{28})$?

$$\newcommand{\Q}{\mathbb{Q}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\al}{\alpha} \newcommand{\bcal}{\mathcal{B}} \newcommand{\qroot}{\sqrt[3]} \newcommand{\froot}{\sqrt[4]...
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20 views

“Non-monic” integral closure?

Let $R\subset S$ be an extension of commutative rings. The integral closure of $R$ in $S$ is $\textrm{IC}(R)=\{s\in S\mid \exists f(x)\in\ R[x]\text{ monic polynomial such that }f(s)=0\}$, which is a ...
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1answer
78 views

$B \subseteq A$ be an integral extension of integral domains where $B$ is normal; if $P \in Spec A$ and $Q=P\cap B$, then $\dim A_P=\dim B_Q$?

Let $B \subseteq A$ be an integral extension of integral domains where $B$ is normal (i.e. integrally closed in its own fraction field ). Let $P$ be a prime ideal of $A$ and $Q=P \cap B$ . Then how to ...
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135 views

In an integral extension $A\subseteq B$, when does the Noetherian ness of $B$ imply that of $A$?

Let $A\subseteq B$ be an integral extension of commutative rings. If $B$ is Noetherian ring and finitely generated as $A$-module, the $A$ is Noetherian ring (we don't even need integral hypothesis ...
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1answer
85 views

Integral domain with Noetherian spectrum and algebraically closed fraction field

If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ? If $R$ is normal (integrally ...
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117 views

Integral domain over which every non-constant irreducible polynomial has degree 1

Let $R$ be an integral domain such that any polynomial $f(X) \in R[X]$ , which is irreducible in $R[X]$, has degree $1$. Then is it true that $R$ is a field ? If this is not true in general , What if ...
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1answer
55 views

For a non-zero principal ideal $I=(x)$ of a ring of integers of an algebraic number field, $|A/I|=| N_{L|\mathbb Q } (x)|$ [duplicate]

Let $L$ be an extension field of $\mathbb Q$ with $[L:\mathbb Q]=n$. Let $A$ be the ring of integers of $L$ i.e. the Integral closure of $\mathbb Z$ in $L$. Then $A$ is a Dedekind domain. If $I$ is a ...
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Given a prime ideal $P$ in a valuation ring $A$, there is a valuation ring $B$ containing $A$ such that $B/PB$ is the fraction field of $A/P$ ?

Let $(A,\mathfrak m)$ be a valuation ring (https://en.wikipedia.org/wiki/Valuation_ring ). Let $P$ be a prime ideal of $A$. I know that $A/P$ and $A_P$ are valuation rings . How to show that there ...
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1answer
62 views

Ideas on proving or disproving a ring is integrally closed

I am working on quotient rings of $\mathbb{Z}[x]$ and I encounter some troubles when there are questions about integrally closed (w.r.t. its field of fraction). For example, when I need to prove (...
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1answer
78 views

Are there any interesting results in quadratic extensions that adjoin $2^k$th roots of unity beyond the Gaussian Integers?

This is admittedly a bit of a broad question I realize, but curiosity has struck me a bit lately. So before you tl;dr; here's the central question: Are there any known generalizable results that arise ...
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1answer
38 views

Irreducible elements remain irreducible in integral closure?

Let $R$ be an integral domain and $\bar R$ be its integral closure in the fraction field. If $b \in R$ is an irreducible element in $R$, then does $b$ remain irreducible in $\bar R$ also ? I can see ...
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1answer
73 views

When prime element in an integral domain stays prime in integral extension

All rings below are assumed to be commutative with unity. Let $A\subseteq B$ be an integral extension of integral domains ($A,B$ are both integral domains and $B$ integral over $A$). If $p\in A$ is ...
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1answer
69 views

In Integral extension of domains injectivity of ring maps from the extension ring followed from the injectivity of the restriction to the base ring.

Let $B$ be an integral ring extension of $A$ and $B$ is an integral domain. Let $f:B \to C$ be a ring homomorphism to any commutative ring $C$ such that the restriction of $f$ to $A$ is injective. ...
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2answers
51 views

Integrality and ring morphisms $f:R\to S$

I am thinking about the following problem: Let $f:R\to S$ be a ring morphism and let $T$ in $R$ be a subring. If $r$ in $R$ is integral over $T$, then it is true that $f(r)$ is integral over the image ...
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1answer
73 views

Ring of integral elements of a commutative ring.

I have been studying the basics of integral closure and as much I have seen (from Atiyah's book) to prove that the integral elements of $A$ in $B$ form a ring we have to use modules and the fact that ...
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128 views

Incomparability Theorem in Extension of rings

Let $A\subset B$ be rings, $B$ is integral over $A$. Let $Q$ and $Q'$ be two prime ideals of $B$ such that $Q\subseteq Q'$ and $Q^c=Q'^c=P$(say). Then prove that $Q=Q'$. Using standard notation we ...
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32 views

$p \subset q$ are two distinct elements of $\mathrm{Spec}(R)$ and $K=Q(R/p),$ then $K$ is not finitely generated as $R$-module.

Suppose $p \subset q$ are two distinct prime ideals of the commutative ring $R.$ Let $K$ be the quotient field of $R/p.$ Show that $K$ is not finitely generated as $R$-module. My guess: If $K$ is ...
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59 views

$R=k[X] , f \in R$ . Let $A=k[X,Y]/(Y^2-f)$ . If $f$ has no square factor , then $A$ is a normal domain

Let $k$ be a field , $R=k[X] , f \in R$ . Let $A=k[X,Y]/(Y^2-f)$ . I know $A$ is an integral domain iff $f$ is not a perfect square and hence forth assume $A$ is a domain. If $f$ has a square factor, ...
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2answers
77 views

Non-flatness of $k[t]$ as a $k[t^2,t^3]$-module

Let $k$ be a field. How to show that $k[t]$ is not flat as a module over $k[t^2,t^3]$ ? Since the ring extension $k[t^2,t^3]\subseteq k[t]$ is integral, it is clear that $k[t]$ is a finitely ...
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1answer
151 views

For non-algebraically closed field $k$ and integer $n>1$, there is a polynomial in $n$-variables over $k$ having only one zero in $k^n$ [duplicate]

Let $k$ be a field which is not algebraically closed and $n>1$ be an integer . Then does there exist $f\in k[X_1,...,X_n]$ such that $Z(f) (:=\{(a_1,...,a_n)\in k^n : f(a_1,...,a_n)=0\}) =\{0\}$ ? ...
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143 views

When is a ring extension satisfying Going-Up and incomparability property , an integral extension?

For an extension of commutative rings $R \subseteq S$ , the extension is said to satisfy Going Up (GU) property if for every chain of prime ideals $P \subseteq P_0$ of $R$ with $P=Q \cap R$ for ...
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62 views

Is $\mathbb{C}[x] \subseteq \mathbb{C}[x, y]/(y^4 + y^3 + y^2 + y + 1)$ an integral ring extension?

I would like to prove or disprove that $S = \mathbb{C}[x, y]/(y^4 + y^3 + y^2 + y + 1)$ is integral over $\mathbb{C}[x]$. Just using the definition of integrality might not be the correct approach, ...
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0answers
68 views

$X\to X/G$ is a finite morphism

Let $G$ be a finite subgroup of automorphisms acting on an affine variety $X\subset\mathbb{A}^n$. I'm trying to prove that the natural projection $\pi:X\to X/G$ is a finite map. First, I'm a bit ...