en>LilHelpa: Typos and general fixes using AWB

2014-06-02T19:55:11Z

Typos and general fixes using AWB

en>Markus Pössel: /* History */

2014-01-20T19:32:08Z

History

← Older revision		Revision as of 21:32, 20 January 2014
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	~~If you are looking for a specific plugin~~, ~~then you can just search for the name~~ of ~~the plugin. Affilo - Theme~~ is ~~the guaranteed mixing~~ of wordpress theme that Mark Ling use for his internet marketing career. This CMS has great flexibility to adapt various extensions and add-ons. After confirming the account, login with your username and password at Ad - Mob. It is found that most of ~~the visitors only look for the results~~ that ~~are displayed on the first page~~ of ~~the search engines and so if you get the service from professional service providers then they strive for the first page ranking~~ of ~~your site and improve the online visibility~~. <br><br>		In [[abstract algebra]], an '''associated prime''' of a [[module (mathematics)\|module]] ''M'' over a [[ring (mathematics)\|ring]] ''R'' is a type of [[prime ideal]] of ''R'' that arises as an [[annihilator (ring theory)\|annihilator]] of a submodule of ''M''. The set of associated primes is usually denoted by <math>\operatorname{Ass}_R(M)\,</math>.

	~~You just download ready made templates~~ to a ~~separate directory~~ and ~~then choose~~ a ~~favorite one in~~ the ~~admin panel~~. If a ~~newbie missed~~ a ~~certain part~~ of the ~~video then they could always rewind~~. ~~This plugin~~ is a ~~must have~~ for ~~anyone who~~ is ~~serious about using Word - Press~~. ~~You can up your site~~'~~s rank~~ with the ~~search engines by simply taking a bit~~ of ~~time with your site~~. ~~For a Wordpress website~~, ~~you don~~'~~t need a powerful web hosting account to host your site~~. <br><br>~~But before choosing any~~ one of ~~these, let~~'~~s compare between the two~~. ~~If you beloved this article and you would like to acquire~~ a ~~lot more info concerning~~ [~~http:~~//~~elkspel~~.~~biz~~/~~profile/breldred~~.~~html backup plugin~~] ~~kindly visit~~ the ~~website~~. ~~Word - Press~~ has ~~different exciting features including~~ a ~~plug~~-~~in architecture with~~ a ~~templating system~~. If ~~Gandhi was empowered with~~ a ~~blogging system~~, ~~every event in his life would have been minutely documented so that it could be recounted~~ to the ~~future generations~~. ~~Every single Theme might be unique~~, ~~providing several alternatives~~ for ~~webpage owners to reap~~ the ~~benefits~~ of ~~in an effort to instantaneously adjust their web page appear~~. ~~If you have any questions on starting~~ a ~~Word - Press food blog or~~ any ~~blog for that matter~~, ~~please post them below and I will try~~ to ~~answer them~~. <br><br>~~You can add keywords but it~~ is ~~best~~ to ~~leave this alone~~. ~~I have compiled~~ a ~~few tips on how you can start~~ a ~~food blog and hopefully the following information and tips can help you~~ to ~~get started on your food blogging creative journey~~. ~~One~~ of the ~~great features~~ of ~~Wordpress is its ability to integrate SEO into your site~~. It'~~s now become~~ a ~~great place~~ to ~~sell it thanks to Woo - Commerce~~. ~~OSDI~~, a ~~Wordpress Development Company based on ahmedabad~~, ~~India~~. <br><br>~~There is no denying that Magento is an ideal platform for building ecommerce websites~~, ~~as it comes~~ with ~~an astounding number~~ of ~~options that can help your online business do extremely well~~. ~~Mahatma Gandhi~~ is ~~known as one~~ of the ~~most prominent personalities and symbols~~ of ~~peace~~, non-~~violence~~ and ~~freedom~~. ~~However~~, ~~you must also manually approve or reject comments so that your website does not promote parasitic behavior. ) Remote Login: With~~ the ~~process~~ of ~~PSD to Wordpress conversion comes~~ the ~~advantage~~ of ~~flexibility~~. ~~For your information~~, it is an ~~open source web content management system~~.		In [[commutative algebra]], associated primes are linked to the [[Lasker–Noether theorem\|Lasker-Noether primary decomposition]] of ideals in commutative [[Noetherian ring]]s. Specifically, if an ideal ''J'' is decomposed as a finite intersection of [[primary ideal]]s, the [[radical of an ideal\|radicals]] of these primary ideals are [[prime ideal]]s, and this set of prime ideals coincides with <math>\operatorname{Ass}_R(R/J)\,</math>.{{sfn\|Lam\|1999\|p=117\|loc=Ex 40B}} Also linked with the concept of "associated primes" of the ideal are the notions of '''isolated primes''' and '''embedded primes'''.

			==Definitions==
			A nonzero ''R'' module ''N'' is called a '''prime module''' if the annihilator <math>\mathrm{Ann}_R(N)=\mathrm{Ann}_R(N')\,</math> for any nonzero submodule ''N' '' of ''N''. For a prime module ''N'', <math>\mathrm{Ann}_R(N)\,</math> is a prime ideal in ''R''.{{sfn\|Lam\|1999\|p=85}}

			An '''associated prime''' of an ''R'' module ''M'' is an ideal of the form <math>\mathrm{Ann}_R(N)\,</math> where ''N'' is a prime submodule of ''M''. In commutative algebra the usual definition is different, but equivalent:{{sfn\|Lam\|1999\|p=86}} if ''R'' is commutative, an associated prime ''P'' of ''M'' is a prime ideal of the form <math>\mathrm{Ann}_R(m)\,</math> for a nonzero element ''m'' of ''M'' or equivalently <math>R/P</math> is isomorphic to a submodule of ''M''.

			In a commutative ring ''R'', minimal elements in <math>\operatorname{Ass}_R(M)</math> (with respect to the set-theoretic inclusion) are called '''isolated primes''' while the rest of the associated primes (i.e., those properly containing associated primes) are called '''embedded prime'''.

			A module is called '''coprimary''' if ''xm'' = 0 for some nonzero ''m'' ∈ ''M'' implies ''x''<sup>''n''</sup>''M'' = 0 for some positive integer ''n''. A nonzero finitely generated module ''M'' over a commutative [[Noetherian ring]] is coprimary if and only if it has exactly one associated prime. A submodule ''N'' of ''M'' is called ''P''-primary if <math>M/N</math> is coprimary with ''P''. An ideal ''I'' is a ''P''-[[primary ideal]] if and only if <math>\operatorname{Ass}_R(R/I) = \{P\}</math>; thus, the notion is a generalization of a primary ideal.

			==Properties==
			Most of these properties and assertions are given in {{harv\|Lam\|2001}} starting on page 86.

			* If ''M' ''⊆''M'', then <math>\mathrm{Ass}_R(M')\subseteq\mathrm{Ass}_R(M)\,</math>. If in addition ''M' '' is an [[essential submodule]] of ''M'', their associated primes coincide.
			* It is possible, even for a commutative local ring, that the set of associated primes of a [[finitely generated module]] is empty. However, in any ring satisfying the [[ascending chain condition]] on ideals (for example, any right or left Noetherian ring) every nonzero module has at least one associated prime.
			* Any [[uniform module]] has either zero or one associated primes, making uniform modules an example of coprimary modules.
			* For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable [[injective module]]s onto the [[spectrum of a ring\|spectrum]] <math>\mathrm{Spec}(R)\,</math>. If ''R'' is an [[Artinian ring]], then this map becomes a bijection.
			*'''Matlis' Theorem''': For a commutative Noetherian ring ''R'', the map from the isomorphism classes of indecomposable injective modules to the spectrum is a bijection. Moreover, a complete set of representatives for those classes is given by <math>E(R/\mathfrak{p})\,</math> where <math>E(-)\,</math> denotes the [[injective hull]] and <math>\mathfrak{p}\,</math> ranges over the prime ideals of ''R''.
			* For a [[Noetherian module]] ''M'' over any ring, there are only finitely many associated primes of ''M''.

			The following properties all refer to a commutative Noetherian ring ''R'':

			* Every ideal ''J'' (through primary decomposition) is expressible as a finite intersection of primary ideals. The radical of each of these ideals is a prime ideal, and these primes are exactly the elements of <math>\mathrm{Ass}_R(R/J)\,</math>. In particular, an ideal ''J'' is a [[primary ideal]] if and only if <math>\mathrm{Ass}_R(R/J)\,</math> has exactly one element.
			* Any [[minimal prime ideal\|prime ideal minimal]] with respect to containing an ideal ''J'' is in <math>\mathrm{Ass}_R(R/J)\,</math>. These primes are precisely the isolated primes.
			* The set theoretic union of the associated primes of ''M'' is exactly the collection of zero-divisors on ''M'', that is, elements ''r'' for which there exists nonzero ''m'' in ''M'' with ''mr'' =0.
			*If ''M'' is a finitely generated module over ''R'', then there is a finite ascending sequence of submodules
			:: <math>0=M_0\subset M_1\subset\cdots\subset M_{n-1}\subset M_n=M\,</math>
			:such that each quotient ''M''<sub>''i''</sub>/''M''<sub>''i−1''</sub> is isomorphic to ''R''/''P''<sub>''i''</sub> for some prime ideals ''P''<sub>''i''</sub>. Moreover every associated prime of ''M'' occurs among the set of primes ''P''<sub>''i''</sub>. (In general not all the ideals ''P''<sub>''i''</sub> are associated primes of ''M''.)
			* Let ''S'' be a multiplicatively closed subset of ''R'' and <math>f: \operatorname{Spec}(S^{-1}R) \to \operatorname{Spec}(R)</math> the canonical map. Then, for a module ''M'' over ''R'',
			*:<math>\operatorname{Ass}_R(S^{-1}M) = f(\operatorname{Ass}_{S^{-1}R}(S^{-1}M)) = \operatorname{Ass}_R(M) \cap \{ P \| P \cap S = \emptyset \}</math>.<ref>{{harvnb\|Matsumura\|1970\|loc=7.C Lemma}}</ref>
			* For a module ''M'' over ''R'', <math>\mathrm{Ass}(M) \subseteq \mathrm{Supp}(M)</math>. Furthermore, the set of minimal elements of <math>\mathrm{Supp}(M)</math> coincides with the set of minimal elements of <math>\mathrm{Ass}(M)</math>. In particular, the equality holds if <math>\mathrm{Ass}(M)</math> consists of maximal ideals.
			* A module ''M'' over ''R'' has [[finite length]] if and only if ''M'' is finitely generated and <math>\mathrm{Ass}(M)</math> consists of maximal ideals.{{citation needed\|date=August 2013}}<!-- or just add a proof? -->

			==Examples==
			*If ''R'' is the ring of integers, then non-trivial [[free abelian group]]s and non-trivial [[abelian group]]s of prime power order are coprimary.
			*If ''R'' is the ring of integers and ''M'' a finite abelian group, then the associated primes of ''M'' are exactly the primes dividing the order of ''M''.
			*The group of order 2 is a quotient of the integers ''Z'' (considered as a free module over itself), but its associated prime ideal (2) is not an associated prime of ''Z''.

			==References==
			{{Reflist}}
			*{{Citation \| last1=Eisenbud \| first1=David \| author1-link=David Eisenbud \| title=Commutative algebra \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics \| isbn=978-0-387-94268-1 \| mr=1322960 \| year=1995 \| volume=150}}
			*{{Citation \| last1=Lam \| first1=Tsit-Yuen \| title=Lectures on modules and rings \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics No. 189 \| isbn=978-0-387-98428-5 \| mr=1653294 \| year=1999}}
			* {{citation \| last1=Matsumura \|first1=Hideyuki \|title=Commutative algebra \|year=1970}}

			[[Category:Commutative algebra]]
			[[Category:Ideals]]
			[[Category:Prime ideals]]
			[[Category:Module theory]]

en>CaballoSinNombre at 22:19, 4 October 2010

2010-10-04T22:19:08Z

New page

If you are looking for a specific plugin, then you can just search for the name of the plugin. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. This CMS has great flexibility to adapt various extensions and add-ons. After confirming the account, login with your username and password at Ad - Mob. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility. 

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en>LilHelpa: Typos and general fixes using AWB

en>Markus Pössel: /* History */

en>CaballoSinNombre at 22:19, 4 October 2010