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	~~In [[mathematics]]~~, ~~an [[Ideal (ring theory)\|ideal]]~~ of ~~a [[commutative ring]] is said~~ to be ~~'''irreducible''' if it cannot~~ be ~~written as~~ the ~~intersection~~ of ~~two larger ideals~~.~~<ref name=~~"~~m98~~">{{citation\|title=Algebraic Geometry\|volume=136\|series=Translations of mathematical monographs\|first=Masayoshi\|last=Miyanishi\|publisher=American Mathematical Society\|year=1998\|isbn=9780821887707\|page=13\|url=http://~~books~~.~~google~~.com/~~books~~?id=~~1reGWSo8XIsC&pg=PA13}}~~.<~~/ref~~>		What exactly is the suitable wedding party? This is a unique as being the pair is that can make up the marriage! No doubt, there are lots of judgements to help make. You may well be stressed by the recommendations accessible for an ideal time. Be sensible and affordable, just like you would for just about any other major determination. Get this per day that may be fondly appreciated. 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	~~Every [[prime ideal]]~~ is ~~irreducible~~.<~~ref~~>~~{{citation\|title=Advanced Algebra\|first=Anthony W~~.~~\|last=Knapp\|series=Cornerstones\|publisher=Springer\|year=2007\|isbn=9780817645229\|page=446\|url=~~http://~~books~~.~~google~~.com/~~books?id=25JfJAgqC8sC&pg=PA446}}~~.<~~/ref~~> ~~Every irreducible ideal~~ of a ~~[[Noetherian ring]] is a [[primary ideal]]~~,<~~ref name="m98"/~~> ~~and consequently for Noetherian rings an irreducible decomposition is~~ a ~~[[primary decomposition]]. Every primary ideal~~ of a ~~[[principal ideal domain]] is an irreducible ideal~~. ~~Every irreducible ideal is~~ a ~~[[primal ideal]]~~.<~~ref~~>~~{{citation~~
	~~\| last = Fuchs \| first = Ladislas~~
	~~\| doi = 10~~.~~2307/2032421~~
	~~\| journal = [[Proceedings of~~ the ~~American Mathematical Society]]~~
	~~\| mr = 0032584~~
	~~\| pages = 1–6~~
	~~\| title = On primal ideals~~
	~~\| volume = 1~~
	~~\| year = 1950}}~~. ~~Theorem 1~~, ~~p. 3~~.<~~/ref~~>

	~~An element of an integral domain is~~ [[~~prime element\|prime]~~] ~~if, and only if, an ideal generated by it is~~ a ~~nonzero prime ideal~~. This is ~~not true~~ for ~~irreducible ideals: an irreducible ideal may~~ be ~~generated by an element that is not an [[irreducible element]], as is the case in~~ <~~math~~>~~\mathbb Z~~<~~/math~~> ~~for~~ the ~~ideal <math>4 \mathbb Z<~~/~~math>: It is not the intersection of two strictly greater ideals~~.

	~~An ideal ''I''~~ of ~~a ring ''A'' can be irreducible only if the~~ [~~[algebraic set~~]~~] it defines is [[irreducible space\|irreducible]] (that is~~, ~~any open subset is dense) for the [[Zariski topology]]~~, or ~~equivalently if~~ the ~~closed space of [[Spectrum of~~ a ~~ring\|spec]] ''A'' consisting of [[prime ideals]] containing ''I'' is irreducible for~~ the ~~[[affine scheme\|spectral topology]]~~.
	~~The converse is~~ not ~~correct, for example~~ the ~~ideal of polynomials in two variables~~ with ~~vanishing terms~~ of ~~first and second order is not irreducible~~.

	~~If '''k''' is an [[algebraically closed field]]~~, ~~choosing the [[radical of an ideal\|radical]] of an irreducible ideal~~ of a ~~polynomial ring over '''k''' is ''~~the ~~same thing'' as choosing an [[embedding]] of the [[affine variety]] of its [[Algebraic geometry\|Nullstelle]]~~ in ~~the affine space~~.

	~~==See also==~~
	* [[irreducible module]]
	* [[irreducible space]]
	* [[Laskerian ring]]

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Ring theory]]~~
	~~[[Category:Algebraic topology]]~~


	~~{{Abstract-algebra-stub}}~~

en>Yobot: /* Definition */clean up, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (8748)

2012-12-05T20:44:25Z

Definition: clean up, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (8748)

← Older revision		Revision as of 22:44, 5 December 2012
Line 1:		Line 1:
	~~Adhering to the Model~~, ~~View, Controller standard, Symfony web development has become the most sought after web development framework that~~ is written ~~in PHP. Many opine that this framework is an out and out winner~~ as ~~it comes in extraordinary open source framework that is free for all~~ the ~~users~~. ~~There are numerous benefits that the features~~ of ~~Symfony Web Development framework brings along~~. ~~This is one platform that provides with numerous development environments and tools too that help in automating the ordinary software engineering processes~~.<br><br>~~There are the code generation tools that are used for prototyping and also used for single click back end administration~~.<br><br>~~The logging features give the administrators detailed information related to the activities~~ of an ~~application~~.~~<br><br>There~~ is a ~~command line interface that helps in automating the deployment of an application between two different servers~~.<br><br>~~There~~ is a ~~debug panel that accelerates the debugging process~~ by ~~way of displaying the information~~ that is ~~needed on~~ the ~~page that~~ the ~~developer is working on.~~<br><br>~~There [http~~:~~//tinyurl~~.~~com/nqbkz6z discount ugg boots~~] is ~~an in built unit as well as a functional testing framework that allows development~~ that is ~~test driven.<br>In short~~, ~~Symfony web development~~ is ~~designed in such a way that it optimizes~~ the ~~web application development and allows an expandable platform for more features to get accommodated with every new release. Generally~~ the ~~complex web applications take~~ a ~~lot~~ of ~~development time~~. ~~Symfony aims at shortening this time and also aims at automating common tasks so that~~ the ~~developer can focus on various specifications~~ of ~~the web applications.<br>Symfony web development framework is a database engine independent framework~~ in ~~PHP and can be very easily installed as well as configured too. It generates a readable code~~ with ~~the help~~ of ~~PHP Web Development. This framework allows integration with various vendor libraries features Propel plugins~~ and ~~also~~ is ~~easily extendable~~. ~~The developers can easily take full advantage of the various built in internalization layer feature which allows data as well as interface translation~~, ~~content localization and reduction in~~ the ~~presentation~~ [~~http://tinyurl.com/nqbkz6z ugg boots sale~~] ~~code amount too by bringing together large shares~~ of ~~the codes through simple function calls. Those who do not have knowledge~~ of ~~the Symfony web development framework can take advantage of the layouts and templates that are built by the HTML designers.<br>It~~ is the ~~built in form generator which offers repopulation and automated validation~~ of the ~~forms. The various applications that are built with Symfony support completely output escaping that protects any kind~~ of ~~attacks through corrupted data. The webmasters too can~~ [~~http://tinyurl.com/nqbkz6z http://tinyurl.com/nqbkz6z~~] ~~benefit greatly with the help of search engine routing, API management features,~~ in ~~built email, smart URLs and Ajax interactions that are easily implementable and also the one line helpers which encapsulate~~ the ~~cross browser compatibility effects of JavaScript~~. ~~There are many~~ [~~http://tinyurl.com/nqbkz6z cheap ugg boots~~] ~~online stops that offer Symfony web development framework as their~~ [~~https://www.gov.uk/search?q~~=~~special+service special service] but one needs to ensure that all the features are offered by the service provider. A look at the past experience in the field will also reveal the kind~~ [~~http~~:~~//tinyurl.com/nqbkz6z discount ugg boots~~] ~~of expertise they have with Symfony web development~~ [~~http~~:~~//tinyurl.com/nqbkz6z cheap ugg boots~~] ~~framework.~~		In [[mathematics]], an [[Ideal (ring theory)\|ideal]] of a [[commutative ring]] is said to be '''irreducible''' if it cannot be written as the intersection of two larger ideals.<ref name="m98">{{citation\|title=Algebraic Geometry\|volume=136\|series=Translations of mathematical monographs\|first=Masayoshi\|last=Miyanishi\|publisher=American Mathematical Society\|year=1998\|isbn=9780821887707\|page=13\|url=http://books.google.com/books?id=1reGWSo8XIsC&pg=PA13}}.</ref>

			Every [[prime ideal]] is irreducible.<ref>{{citation\|title=Advanced Algebra\|first=Anthony W.\|last=Knapp\|series=Cornerstones\|publisher=Springer\|year=2007\|isbn=9780817645229\|page=446\|url=http://books.google.com/books?id=25JfJAgqC8sC&pg=PA446}}.</ref> Every irreducible ideal of a [[Noetherian ring]] is a [[primary ideal]],<ref name="m98"/> and consequently for Noetherian rings an irreducible decomposition is a [[primary decomposition]]. Every primary ideal of a [[principal ideal domain]] is an irreducible ideal. Every irreducible ideal is a [[primal ideal]].<ref>{{citation
			\| last = Fuchs \| first = Ladislas
			\| doi = 10.2307/2032421
			\| journal = [[Proceedings of the American Mathematical Society]]
			\| mr = 0032584
			\| pages = 1–6
			\| title = On primal ideals
			\| volume = 1
			\| year = 1950}}. Theorem 1, p. 3.</ref>

			An element of an integral domain is [[prime element\|prime]] if, and only if, an ideal generated by it is a nonzero prime ideal. This is not true for irreducible ideals: an irreducible ideal may be generated by an element that is not an [[irreducible element]], as is the case in <math>\mathbb Z</math> for the ideal <math>4 \mathbb Z</math>: It is not the intersection of two strictly greater ideals.

			An ideal ''I'' of a ring ''A'' can be irreducible only if the [[algebraic set]] it defines is [[irreducible space\|irreducible]] (that is, any open subset is dense) for the [[Zariski topology]], or equivalently if the closed space of [[Spectrum of a ring\|spec]] ''A'' consisting of [[prime ideals]] containing ''I'' is irreducible for the [[affine scheme\|spectral topology]].
			The converse is not correct, for example the ideal of polynomials in two variables with vanishing terms of first and second order is not irreducible.

			If '''k''' is an [[algebraically closed field]], choosing the [[radical of an ideal\|radical]] of an irreducible ideal of a polynomial ring over '''k''' is ''the same thing'' as choosing an [[embedding]] of the [[affine variety]] of its [[Algebraic geometry\|Nullstelle]] in the affine space.

			==See also==
			* [[irreducible module]]
			* [[irreducible space]]
			* [[Laskerian ring]]

			==References==
			{{reflist}}

			[[Category:Ring theory]]
			[[Category:Algebraic topology]]


			{{Abstract-algebra-stub}}

en>R.e.b.: Adding/removing wikilink(s)

2012-03-15T20:53:11Z

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