Continuous linear extension - Revision history

en>Monkbot: /* References */Task 3: Fix CS1 deprecated coauthor parameter errors

2014-05-07T01:41:38Z

References: Task 3: Fix CS1 deprecated coauthor parameter errors

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	~~In [[mathematics]], in particular [[commutative algebra]], the concept~~ of ~~'''fractional ideal''' is introduced in~~ the ~~context of [[integral domain]]s and~~ is ~~particularly fruitful in the study~~ of ~~[[Dedekind domain]]s~~. ~~In some sense, fractional ideals~~ of ~~an integral domain are like ideals where [[denominator]]s are allowed. In contexts where fractional ideals and ordinary [[ring ideal]]s are both under discussion~~, ~~the latter are sometimes termed '''''integral ideals''''' for clarity~~.		It is non-contagious genetically inherited medical condition. Are you currently considering finding a reasonable and powerful way of whitening the skin that is devoid of any harmful side effects. One study suggests a novel method of testing multiple topical skin-lightening methods prior to initiating therapy, using UV-induced skin tanning. health and preventing diseases such as cancer and coronary heart disease. Yoga and other forms of exercise can reduce stress, and make you healthier. <br><br>Bleaching cream can also reduce scars not just your skin pigmentation. And many hyperhidrosis sufferers (including myself) will vouch that it can completely consume and ruin any situation. Try and look for a product which contains Alpha Arbutin, as this substance has had a very high success rate when whitening a variety of skin tones. However, it would usually take at least six months before results are seen. It not only gradually whitens skin naturally, but your skin also becomes brighter and healthier without any harmful side effects. <br><br>It has a high success rate on patients in regards to decreasing the amount of skin oil produced, but there is a nasty side effect that leaves many users of the medication with depression. Using layers and selection tools may seem more like work than fun to some, but if you really want to experience the full power of Paint - Shop Pro, it's a good idea to learn as much about these features as you can. Cigarettes can make your teeth turn yellow, but with the treatment, you can forget all about it and turn your smile into a brand new one. According to these colors and to the type of skin you have certain products are recommended and certain are prohibited. They work simply by ramping way up mobile or portable return costs. <br><br>When 15 minutes have passed, you must remove the mask with warm water and soap and dry with a soft towel. Other skin conditions that can occasionally be confused for age areas are moles, actinic keratosis, lentigo maligna and melanoma. This Photoshop tutorial explains how to gradually change the skin tone of a subject in a photograph or other image using Photoshop's eyedropper, lasso, and paint bucket tools. It usually lasts approximately six month, depending on the patient’s diet and daily habits, which can affect longevity. Fresh produce will provide you with the right amount of energy while supplying crucial vitamins and minerals to keep your mind and body healthy. <br><br>Regardless of whether the allergen in question is of the animal, fruit or vegetable variety, just omit the offending product, leaving you to savor one hundred percent safe and homemade method to whiten the skin. After makeing a paste with these ingredients, apply it on your skin. Some over the counter face whitening creams can cause damage to your face if misused. There are several remedies, special balms aim at skin bleaching for those who find darkness firmness as a result of sun tan, age group or smog. It is said that skin is the mirror of our inner body.<br><br>In case you have almost any questions about exactly where in addition to the best way to make use of [http://stylax.info/ how to whiten your skin], you can e-mail us on our web-site.

	~~==Definition~~ and ~~basic results==~~

	~~Let ''R'' be an [[integral domain]],~~ and ~~let ''K'' be its [[field of fractions]]~~. ~~A '''fractional ideal''' of ''R'' is an ''R''-[[submodule]] ''I''~~ of ~~''K'' such that there exists a non-zero ''r'' ∈ ''R'' such that ''rI'' ⊆ ''R''. The element ''r''~~ can be thought of as clearing out the denominators in ''I''. The '''principal fractional ideals''' are those ''R''-submodules of ''K'' generated by a single nonzero element of ''K''. A fractional ideal ''I'' is contained in ''R'' if, and ~~only if, it is an ('integral') ideal of ''R''~~.

	~~A fractional ideal ''I'' is called '''invertible''' if there is another fractional ideal ''J'' such that ''IJ'' = ''R'' (where ''IJ'' = { a~~<~~sub~~>1<~~/sub~~>~~b<sub>1</sub> +~~ a~~<sub>2</sub>b<sub>2</sub> +~~ ... ~~+ a<sub>''n''</sub>b~~<~~sub~~>~~''n''~~<~~/sub~~> : a~~<sub>''i''</sub> ∈ ''I''~~, ~~b<sub>''i''</sub> ∈ ''J'', ''n'' ∈ '''Z'''<sub>>0</sub> }~~ is ~~called the '''product'''~~ of the ~~two fractional ideals)~~. ~~The set of invertible fractional ideals form an [[abelian group]] with respect~~ to ~~above product~~, ~~where the identity is the [[unit ideal]] ''R'' itself. This group is called~~ the ~~'''group~~ of ~~fractional ideals~~'~~'' of ''R''. The principal fractional ideals form~~ a ~~subgroup~~. ~~A (nonzero) fractional ideal is invertible if~~, ~~and only if~~, it ~~is [[projective module\|projective]] as an ''R''-module~~.

	~~Every [[finitely generated module\|finitely generated ''R''-submodule~~ of ~~''K'']] is a fractional ideal~~ and ~~if ''R'' is [[Noetherian ring\|noetherian]] these~~ are ~~all the fractional ideals of ''R''~~.

	~~==Dedekind domains==~~

	~~In Dedekind domains~~, the ~~situation is much simpler~~. ~~In particular~~, ~~every non-zero fractional ideal is invertible~~. ~~In fact~~, ~~this property characterizes Dedekind domains: an integral domain is a Dedekind domain if~~, and ~~only if~~, ~~every non-zero fractional ideal is invertible~~.

	~~The [[quotient group]] of fractional ideals by~~ the ~~subgroup~~ of ~~principal fractional ideals is an important invariant of a Dedekind domain called the [[ideal class group]]~~.

	==<~~span id="divisorial"~~><~~/span~~>~~Divisorial ideal==~~
	~~Let <math>\tilde I</math> denote~~ the ~~intersection~~ of ~~all principal fractional ideals containing~~ a ~~nonzero fractional ideal ''I''. Equivalently~~,
	~~:<math>\tilde I = (R : (R : I)),</math>~~
	~~where~~
	~~:<math>(R : I) = \{ x \in K : xI \subseteq R \} </math> ([[ideal quotient]])~~
	~~If <math>\tilde I = I</math> then ''I'' is called '''divisorial'''~~.~~<ref>{{harvnb\|Bourbaki\|1998\|loc=§VII~~.~~1}}</ref> In other words~~, a ~~divisorial ideal is a nonzero intersection~~ of ~~some nonempty set of fractional principal ideals~~. ~~If ''I''~~ is ~~divisorial and ''J'' is a nonzero fractional ideal, then (''I'' : ''J'')~~ is ~~divisorial.~~

	~~An integral domain that satisfies~~ the ~~ascending chain conditions on divisorial ideals is called a [[Mori domain]]~~.<~~ref~~>http://~~projecteuclid.org/DPubS/Repository/1~~.0/~~Disseminate?view=body&id=pdffirstpage_1&handle=euclid.rmjm/1187453107</ref>~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*Chapter 9 of {{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| last2=Macdonald \| first2=I.G. \| author2-link=Ian G. Macdonald \| title=Introduction to ~~Commutative Algebra \| publisher=Westview Press \| isbn=978~~-~~0-201~~-~~40751-8 \| year=1994}}~~
	*Chapter VII.~~1 of {{Citation \| last1=Bourbaki \| first1=Nicolas \| author1-link=Nicolas Bourbaki \| title=Commutative algebra \| publisher=[[Springer Verlag]] \| edition=2nd \| year=1998 \| isbn=3-540-64239-0 }}~~
	*Chapter 11 of {{Citation \| last1=Matsumura \| first1=Hideyuki \| title=Commutative ring theory \| publisher=[[Cambridge University Press]] \| edition=2nd \| series=Cambridge Studies in Advanced Mathematics \| isbn=978-0-521-36764-6 \| mr=1011461 \| year=1989 \| volume=8}}

	~~{{DEFAULTSORT:Fractional Ideal}}~~
	~~[[Category:Ideals]]~~
	~~[[Category:Algebraic number theory]]~~

96.235.189.151: /* Theorem */

2014-02-04T03:40:54Z

Theorem

← Older revision		Revision as of 05:40, 4 February 2014
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			In [[mathematics]], in particular [[commutative algebra]], the concept of '''fractional ideal''' is introduced in the context of [[integral domain]]s and is particularly fruitful in the study of [[Dedekind domain]]s. In some sense, fractional ideals of an integral domain are like ideals where [[denominator]]s are allowed. In contexts where fractional ideals and ordinary [[ring ideal]]s are both under discussion, the latter are sometimes termed '''''integral ideals''''' for clarity.

			==Definition and basic results==

	~~To one~~'~~s offense~~, ~~you might experience Gunboats which~~ can ~~simply shoot at enemy rights coming from~~ a ~~prolonged periods~~ of ~~time range and Landing Crafts which you must seal when you train gadgets for example Rifleman~~, ~~Heavy, Zooka, Warrior~~ and ~~Reservoirs. To your village defenses~~, ~~you might have structures like Mortar, Products Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Drive Launcher to assist you eradicate enemies~~.<br><br>~~To appreciate coins and gems, kind obtain the Clash amongst Clans hack equipment by- clicking on the purchase button~~. ~~Contingent towards the operating framework that you are utilizing, you will operate the downloaded document as the admin~~. ~~If you adored this write-up and you would like to receive additional facts concerning [http~~://~~circuspartypanama.com clash of clans bot] kindly visit the web page. Furnish a person~~'~~s log in Id and select the gadget. When this~~, ~~you are to get in the quantity of diamond jewelry or coins that if you want to and start off unquestionably the Clash of Clans get into instrument.~~<br><br>~~Conclusion There are a regarding Apple fans who within~~ the above ~~game all on~~ the ~~planet~~. This ~~generation has donrrrt been~~ the ~~JRPG~~'~~s best; in fact it~~'~~s been recently unanimously its worst~~. ~~Exclusively at Target: Mission: Impossible 4-Pack DVD Set with all 4 Mission: Impossible movies~~)~~. Although it~~ is ~~a special day's grand gifts~~ and ~~gestures~~, ~~one Valentines Day~~ is ~~likely to blend into another far too easily~~. ~~clash~~ of ~~clans~~ is ~~among~~ the ~~quickest rising video gaming as~~ of ~~late~~.~~<br><br>Be charged attention to how money your teenager~~ is ~~considered spending on video console games~~. ~~These products are certainly cheap~~ and ~~there~~ is ~~without question often~~ the ~~option~~ of a ~~buying more add-ons in just~~ the ~~game itself~~. ~~Set monthly and yearly limits on~~ the ~~quantity~~ of ~~of money that may be spent on video games~~. ~~Also~~, ~~need [https~~://~~www.flickr.com/search/?q=conversations conversations~~] ~~with your kids about budgeting.~~<br><br>~~Exclusive some online games provde the comfort of creating a true-entire world call accessible in the movie game itself. This~~ is ~~usually a downside so as to full-monitor game titles. You don~~'t want these kind of people using up even a good deal more of your time and after that energy than within your main budget place a days clock of your in close proximity to to your display movie screen to be able if you want to monitor just how very long you'~~ve been enjoying~~.<br><br>~~Conserve lots~~ of some ~~money on your main games~~, ~~think about opt-in into a assistance~~ that ~~you just can rent payments programs from. The price of these lease plans for~~ the ~~year~~ is ~~regarded as normally under the rate of two video games. You can preserve the game titles until you do more than them and simply email out them back~~ a ~~lot more and purchase another one particular particular~~.<br><br>~~The actual leap into the pre~~-~~owned or operated xbox game marketplace~~. ~~Several online poker players will get a Battle of Clans Hack and complete this game really fast~~. ~~Several shops let these games being dealt in just~~ to ~~promote them at the very lessened cost~~. ~~Might be by far one~~ of ~~the most cost~~-~~effective technique to be newer video games absolutely no higher cost.~~		Let ''R'' be an [[integral domain]], and let ''K'' be its [[field of fractions]]. A '''fractional ideal''' of ''R'' is an ''R''-[[submodule]] ''I'' of ''K'' such that there exists a non-zero ''r'' ∈ ''R'' such that ''rI'' ⊆ ''R''. The element ''r'' can be thought of as clearing out the denominators in ''I''. The '''principal fractional ideals''' are those ''R''-submodules of ''K'' generated by a single nonzero element of ''K''. A fractional ideal ''I'' is contained in ''R'' if, and only if, it is an ('integral') ideal of ''R''.

			A fractional ideal ''I'' is called '''invertible''' if there is another fractional ideal ''J'' such that ''IJ'' = ''R'' (where ''IJ'' = { a<sub>1</sub>b<sub>1</sub> + a<sub>2</sub>b<sub>2</sub> + ... + a<sub>''n''</sub>b<sub>''n''</sub> : a<sub>''i''</sub> ∈ ''I'', b<sub>''i''</sub> ∈ ''J'', ''n'' ∈ '''Z'''<sub>>0</sub> } is called the '''product''' of the two fractional ideals). The set of invertible fractional ideals form an [[abelian group]] with respect to above product, where the identity is the [[unit ideal]] ''R'' itself. This group is called the '''group of fractional ideals''' of ''R''. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is [[projective module\|projective]] as an ''R''-module.

			Every [[finitely generated module\|finitely generated ''R''-submodule of ''K'']] is a fractional ideal and if ''R'' is [[Noetherian ring\|noetherian]] these are all the fractional ideals of ''R''.

			==Dedekind domains==

			In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains: an integral domain is a Dedekind domain if, and only if, every non-zero fractional ideal is invertible.

			The [[quotient group]] of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the [[ideal class group]].

			==<span id="divisorial"></span>Divisorial ideal==
			Let <math>\tilde I</math> denote the intersection of all principal fractional ideals containing a nonzero fractional ideal ''I''. Equivalently,
			:<math>\tilde I = (R : (R : I)),</math>
			where
			:<math>(R : I) = \{ x \in K : xI \subseteq R \} </math> ([[ideal quotient]])
			If <math>\tilde I = I</math> then ''I'' is called '''divisorial'''.<ref>{{harvnb\|Bourbaki\|1998\|loc=§VII.1}}</ref> In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals. If ''I'' is divisorial and ''J'' is a nonzero fractional ideal, then (''I'' : ''J'') is divisorial.

			An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a [[Mori domain]].<ref>http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdffirstpage_1&handle=euclid.rmjm/1187453107</ref>

			==Notes==
			{{reflist}}

			==References==
			*Chapter 9 of {{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| last2=Macdonald \| first2=I.G. \| author2-link=Ian G. Macdonald \| title=Introduction to Commutative Algebra \| publisher=Westview Press \| isbn=978-0-201-40751-8 \| year=1994}}
			*Chapter VII.1 of {{Citation \| last1=Bourbaki \| first1=Nicolas \| author1-link=Nicolas Bourbaki \| title=Commutative algebra \| publisher=[[Springer Verlag]] \| edition=2nd \| year=1998 \| isbn=3-540-64239-0 }}
			*Chapter 11 of {{Citation \| last1=Matsumura \| first1=Hideyuki \| title=Commutative ring theory \| publisher=[[Cambridge University Press]] \| edition=2nd \| series=Cambridge Studies in Advanced Mathematics \| isbn=978-0-521-36764-6 \| mr=1011461 \| year=1989 \| volume=8}}

			{{DEFAULTSORT:Fractional Ideal}}
			[[Category:Ideals]]
			[[Category:Algebraic number theory]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-09T00:11:36Z

ISBNs (Build KE)

New page

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