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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.
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==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]]

Latest revision as of 02:41, 7 May 2014

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.

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.

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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.

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.

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