# Questions tagged [integral-dependence]

Integral dependence (also known as algebraic dependence) is a condition on ring extensions $R\subseteq S$: we say $s\in S$ is integral over $R$ if there is some $f(x) \in R[x]$ such that $f(s)=0$.

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### Integral Dependence

A module theoretical equivalence of integral dependence states as follows: Let $S$ be a ring and $R \subseteq S$ be a subring, and $1\in R$. Then $x \in S$ is integral over $R$ if and only if $R[x]$ ...
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### A question about integral closure of a finitely generated $\Bbb{Z}$-algebra

Suppose $R$ is a finitely generated $\mathbb{Z}$-algebra and also an integral domain, $\dim R=1$. Is A (the integral closure of $R$ in $Q(R)$ (quotient field of R)) a finitely generated $\mathbb{Z}$-...
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### Ring of fractions of an integrally closed integral domain is also integrally closed

I hope that the following questions can be solved by Ring Theory concepts as I'm not studying further yet. Let $A$ be an integral domain and $K$ its field of fractions. An integral domain $A$ is said ...
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### Extending an automorphism to the integral closure

I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following. Let $\sigma$ be an automorphism of ...
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### Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
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### Is the preimage of the non-normal locus a divisor?

Let $X$ be a complex, affine variety. Let $\nu:\tilde X\to X$ be the normalization of $X$ and denote by $D\subseteq X$ the closed set of points where $\nu$ fails to be an isomorphism, i.e. $D$ is the ...
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### Which type of correlation should I use?

I am beginner in statistics. I have excel table with few columns. I would like to find correlation between the variables. I have to make an essay to my boss and he wants concrete answers. I searchin ...
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### Extension of Integral Domains

Let $S\subset R$ be an extension of integral domains. If the ideal $(S:R)=\{s\in S\mid sR\subseteq S\}$ is finitely generated, show that $R$ is integral over $S$. My first attempt was to show that $R$...
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### $F/E$ a finite Galois extension, the integral closure of $E[X]$ in $F(X)$

If $F/E$ is a finite Galois extension, then one can show that $F(X)/E(X)$ is also a finite Galois extension of the same degree (a basis for $F/E$ is also a basis for $F(X)/E(X)$). Since $E[X]$ is a ...
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### Integral extension and field

I came a cross a question that I don't know how to solve Problem: $A,B$ are commutative domains and $A\subseteq B$. Show if that $B$ is a field and every element of $B$ is the root of a non-trivial ...
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### Are locally integral extensions integral? [closed]

Let $R$ be a commutative ring with unit and let $A \subseteq B$ be commutative $R$-algebras. Suppose $B_P$ is integral over $A_P$ for all primes $P$ of $R$. Is $B$ then necessarily integral over $A$ ?
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### Proposition 5.15 from Atiyah and Macdonald: Integral Closure and Minimal Polynomial

I am having some trouble understanding Proposition 5.15 in Introduction to Commutative Algebra by Atiyah and Macdonald. Let $A\subset B$ be integral domains, $A$ integrally closed, and $x\in B$ be ...
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### Difference between algebraic and integral extension

I have been reading Miles Reid Undergraduate Commutative Algebra and in chapter 4 he talks about a crucial difference between algebraic extension and integral extension (see the picture below). Now I ...
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### Integral dependence and field extension

Let $R$ be a domain (commutative with unity). $k$ is field algebraically dependent on $k_0$. $A$ is some ideal of $R \otimes_{k_0} k$ and $A_0$ = $A \cap R$. How to prove that $(R \otimes_{k_0} k)/A$ ...
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### Normalization of a quotient ring of polynomial rings (Reid, Exercise 4.6)

I solved all parts of Exercise 4.6 of the book Undergraduate Commutative Algebra of Miles Reid except the last one. Let $A=k[X]$ and $f\in A$ has a square factor but it is not a square polynomial ...
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### If $x$ is integral over $A_m$ for all maximal ideals $m$, then $x$ is integral over $A$
I am going over an old exam, and there is this question that I am stuck: Given $A$ a commutative ring with unity, show that if $x\in\operatorname{Frac}(A)$ is integral over $A_m$ for all maximal ...
### Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then $x$ is not integral over $S$
Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then I want to show that $x$ is not integral over $S$ I'm not seeing why this is the case....