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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Latest revision as of 18:02, 9 September 2008
Let 
be a ring and 
a subring of . We say that an element 
is integral over if there is a monic polynomial 
, with each 
so that 
. The elements of that are integral over form a subring 
of which contains . We call the integral closure of in . If 
, then we say that is integral over . If 
, then we say that is integrally closed in . If is integrally closed in its field of fractions, then we call it integrally closed.
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