Difference between revisions of "Integral closure"

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Let <math>S</math> be a [[ring]] and <math>R</math> a subring of <math>S</math>. We say that an element <math>s\in S</math> is '''integral''' over <math>R</math> if there is a [[monic polynomial]] <math>f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0</math>, with each <math>a_i\in R</math> so that <math>f(s)=0</math>. The elements of <math>S</math> that are integral over <math>R</math> form a subring <math>T</math> of <math>S</math> which contains <math>R</math>. We call <math>T</math> the '''integral closure''' of <math>R</math> in <math>S</math>. If <math>T=S</math>, then we say that <math>S</math> is '''integral''' over <math>R</math>. If <math>T=R</math>, then we say that <math>R</math> is '''integrally closed''' in <math>S</math>. If <math>R</math> is integrally closed in its [[field of fractions]], then we call it '''integrally closed'''.
 
Let <math>S</math> be a [[ring]] and <math>R</math> a subring of <math>S</math>. We say that an element <math>s\in S</math> is '''integral''' over <math>R</math> if there is a [[monic polynomial]] <math>f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0</math>, with each <math>a_i\in R</math> so that <math>f(s)=0</math>. The elements of <math>S</math> that are integral over <math>R</math> form a subring <math>T</math> of <math>S</math> which contains <math>R</math>. We call <math>T</math> the '''integral closure''' of <math>R</math> in <math>S</math>. If <math>T=S</math>, then we say that <math>S</math> is '''integral''' over <math>R</math>. If <math>T=R</math>, then we say that <math>R</math> is '''integrally closed''' in <math>S</math>. If <math>R</math> is integrally closed in its [[field of fractions]], then we call it '''integrally closed'''.
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[[Category:Ring theory]]
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Latest revision as of 18:02, 9 September 2008

Let $S$ be a ring and $R$ a subring of $S$. We say that an element $s\in S$ is integral over $R$ if there is a monic polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$, with each $a_i\in R$ so that $f(s)=0$. The elements of $S$ that are integral over $R$ form a subring $T$ of $S$ which contains $R$. We call $T$ the integral closure of $R$ in $S$. If $T=S$, then we say that $S$ is integral over $R$. If $T=R$, then we say that $R$ is integrally closed in $S$. If $R$ is integrally closed in its field of fractions, then we call it integrally closed.

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