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en>Xezbeth: Unlinked: Continuum

2015-01-12T08:48:35Z

← Older revision		Revision as of 10:48, 12 January 2015
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	~~They~~ call me ~~Emilia~~. ~~Hiring~~ is his ~~profession~~. ~~One~~ of the issues ~~she enjoys~~ most ~~is to do aerobics and now she is trying to make money with it. South Dakota is where I've always been residing~~.<br><br>~~Feel free to surf to~~ my ~~web blog -~~ [http://~~www~~.~~gaysphere~~.~~net~~/blog/~~122921 home std test kit~~]		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.<br><br>Here is my page :: over the counter std test ([http://ayf.csc4rbg.org/blog/view/97969/prevent-candidiasis-before-hand-with-these-tips please click the next page])

142.3.90.226 at 07:29, 23 February 2014

2014-02-23T07:29:33Z

← Older revision		Revision as of 09:29, 23 February 2014
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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?-->~~		They call me Emilia. Hiring is his profession. One of the issues she enjoys most is to do aerobics and now she is trying to make money with it. South Dakota is where I've always been residing.<br><br>Feel free to surf to my web blog - [http://www.gaysphere.net/blog/122921 home std test kit]

	~~==Commutative rings==~~
	~~{{main\|Integrally closed domain}}~~

	~~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]]~~.

	~~If the ring is not a domain, typically being integrally closed means that every [[local ring]] is an integrally closed domain.~~

	~~Sometimes a domain that is integrally closed is called "normal" if it is integrally closed and being thought of as a variety.~~
	~~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.~~

	~~==Ordered groups==~~

	~~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.~~

	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.
	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.

	~~==References==~~
	* R. Hartshorne, ''Algebraic Geometry'', Springer-Verlag (1977)
	* M. Atiyah, I. Macdonald ''Introduction to commutative algebra'' Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
	* H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
	* A.M.W Glass, ''Partially Ordered Groups'', World Scientific, 1999

	~~[[Category:Ordered groups]]~~
	~~[[Category:Commutative algebra]~~]

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q5165895

2013-03-13T20:54:38Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q5165895

← Older revision		Revision as of 22:54, 13 March 2013
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	~~The title~~ of the ~~writer~~ is ~~Numbers~~. ~~She~~ is ~~a librarian but she~~'~~s always needed her own business~~. ~~For~~ a ~~whilst she's been~~ in ~~South Dakota~~. ~~To collect cash~~ is 1 of the ~~issues I love most~~.<br><br>~~Here~~ is ~~my web blog~~ ... [~~http~~:~~//Tinyurl.com/k7cuceb http~~:~~//Tinyurl.com/k7cuceb~~]		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?-->

			==Commutative rings==
			{{main\|Integrally closed domain}}

			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]].

			If the ring is not a domain, typically being integrally closed means that every [[local ring]] is an integrally closed domain.

			Sometimes a domain that is integrally closed is called "normal" if it is integrally closed and being thought of as a variety.
			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.

			==Ordered groups==

			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.

			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.
			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.

			==References==
			* R. Hartshorne, ''Algebraic Geometry'', Springer-Verlag (1977)
			* M. Atiyah, I. Macdonald ''Introduction to commutative algebra'' Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
			* H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
			* A.M.W Glass, ''Partially Ordered Groups'', World Scientific, 1999

			[[Category:Ordered groups]]
			[[Category:Commutative algebra]]

en>Crowsnest: /* Substantive derivative */ move sentence-ending dot to correct position

2012-04-27T14:18:53Z

Substantive derivative: move sentence-ending dot to correct position

New page

The title of the writer is Numbers. She is a librarian but she's always needed her own business. For a whilst she's been in South Dakota. To collect cash is 1 of the issues I love most.<br><br>Here is my web blog ... [http://Tinyurl.com/k7cuceb http://Tinyurl.com/k7cuceb]