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| In [[mathematics]], more specifically in [[abstract algebra]], the concept of '''integrally closed''' has two meanings, one for [[group (mathematics)|groups]] and one for [[ring (mathematics)|rings]]. <!-- are the concepts related?-->
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| ==Commutative rings==
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| {{main|Integrally closed domain}}
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| A commutative ring <math>R</math> contained in a ring <math>S</math> is said to be '''integrally closed''' in <math>S</math> if <math>R</math> is equal to the [[integral closure]] of <math>R</math> in <math>S</math>. That is, for every monic polynomial ''f'' with coefficients in <math>R</math>, every root of ''f'' belonging to ''S'' also belongs to <math>R</math>. Typically if one refers to a domain being integrally closed without reference to an [[overring]], it is meant that the ring is integrally closed in its [[field of fractions]].
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| If the ring is not a domain, typically being integrally closed means that every [[local ring]] is an integrally closed domain.
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| Sometimes a domain that is integrally closed is called "normal" if it is integrally closed and being thought of as a variety.
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| In this respect, the normalization of a [[Algebraic variety|variety]] (or [[scheme (mathematics)|scheme]]) is simply the <math>\operatorname{Spec}</math> of the integral closure of all of the rings.
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| ==Ordered groups==
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| An [[ordered group]] ''G'' is called '''integrally closed''' [[if and only if]] for all elements ''a'' and ''b'' of ''G'', if ''a''<sup>''n''</sup> ≤ ''b'' for all natural ''n'' then ''a'' ≤ 1.
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| This property is somewhat stronger than the fact that an ordered group is [[Archimedean property|Archimedean]]. Though for a [[lattice-ordered group]] to be integrally closed and to be Archimedean is equivalent.
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| We have the surprising theorem that every integrally closed [[directed set|directed]] group is already [[abelian group|abelian]]. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed. Furthermore, every archimedean lattice-ordered group is abelian.
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| ==References==
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| * R. Hartshorne, ''Algebraic Geometry'', Springer-Verlag (1977)
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| * M. Atiyah, I. Macdonald ''Introduction to commutative algebra'' Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
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| * H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
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| * A.M.W Glass, ''Partially Ordered Groups'', World Scientific, 1999
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| [[Category:Ordered groups]]
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| [[Category:Commutative algebra]]
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Oscar is what my wife enjoys to call me and I completely dig that title. Years in the past we moved to North Dakota. He used to be unemployed but now he is a computer operator but his marketing never comes. Body developing is one of the issues I adore most.
Here is my page :: over the counter std test (please click the next page)