Quantity adjustment - Revision history

208.50.124.65: ce: fixing "is where" construction

2014-08-24T23:46:44Z

ce: fixing "is where" construction

← Older revision		Revision as of 01:46, 25 August 2014
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	~~{{Refimprove\|date=September 2011}}~~		Greetings. Let me begin by telling you the author's name - Phebe. Puerto Rico is where he's usually been living but she requirements to transfer simply because of her family. One of the things she enjoys most is to read comics and she'll be beginning something else alongside with it. I am a meter reader but I strategy on changing it.<br><br>my web page - [http://www.animecontent.com/blog/276650 over the counter std test]
	~~In [[abstract algebra]], a [[module (mathematics)\|module]] is '''indecomposable'~~'~~' if it is non-zero and cannot be written as a [[direct sum of modules\|direct sum]] of two non-zero [[submodule]]~~s~~.<ref~~ name~~="Jacobson"> Jacobson (2009), p. 111.</ref>~~

	~~Indecomposable is a weaker notion than [[simple module]]:~~
	~~simple means "no proper submodule" <math>N < M</math>,~~
	~~while indecomposable "not expressible as <math>N \oplus P = M</math>"~~.

	~~A direct sum of indecomposables~~ is ~~called ''~~'~~completely decomposable'''; this is weaker than being [[semisimple module\|semisimple]], which is a direct sum of [[simple module]]~~s.

	~~==Motivation==~~
	~~In many situations, all modules of interest are completely decomposable; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need~~ to ~~be studied. This is the case for modules over a~~
	~~[[field (mathematics)\|field]] or [[principal ideal domain\|PID]],~~
	~~and underlies [[Jordan normal form]]~~ of ~~[[linear operator\|operators]].~~

	~~==Examples==~~
	~~=== Field ===~~
	Modules over [[field (mathematics)\|field]]s are [[vector space]]s. A vector space is indecomposable if and only if its [[dimension (linear algebra)\|dimension]] is 1. So every vector space is completely decomposable (indeed, semisimple), with infinitely many summands if the dimension is infinite.<ref> Jacobson (2009), p. ~~111, in comments after Prop. 3.1.</ref>~~

	~~===PID===~~
	~~Finitely-generated modules over [[principal ideal domain\|PIDs]] are classified by the~~
	~~[[structure theorem for finitely generated modules over a principal ideal domain]]:~~
	~~the primary decomposition is a decomposition into indecomposable modules,~~
	~~so every finitely-generated module over a PID is completely decomposable.~~

	~~Explicitly, the modules~~ of the ~~form <math>R/p^n</math> for [[prime ideal]]s ''p'' (including ''p''=0, which yields ''R'') are indecomposable. Every finitely-generated ''R''-module~~ is ~~a direct sum of these. Note that this is simple if~~ and ~~only if~~ '~~'n''=1 (or ''p''=0); for example, the cyclic group of order 4, '''Z'''/4, is indecomposable but not simple –~~ it ~~has the subgroup 2'''Z'''/4 of order 2,~~ but ~~this does not have a complement.~~

	~~Over the [[integer]]s '''Z''', modules are [[abelian group]]s. A finitely-generated abelian group is indecomposable if and only if~~ it is [[isomorphic]] to '''Z''' or to a [[factor group]] of the form <math>\mathbf{Z}/p^n\mathbf{Z}</math> for some [[prime number]] ''p'' and some positive integer ''n''. Every [[finitely-generated abelian group]] is a direct sum of (finitely many) indecomposable abelian groups.

	~~There are, however, other indecomposable abelian groups which are not finitely generated; the [[rational number]]s '''Q''' form the simplest example.~~

	~~For a fixed positive integer ''n'', consider the ring ''R'' of ''n''-by-''n'' [[matrix (mathematics)\|matrices]] with entries from the [[real number]]s (or from any other field ''K'')~~. ~~Then ''K''~~<~~sup~~>~~''n''~~<~~/sup~~> ~~is a left ''R''~~-~~module (the scalar multiplication is~~ [[matrix multiplication]]). This is [[up to isomorphism]] the only indecomposable module over ''R''. Every left ''R''-module is a direct sum of (finitely or infinitely many) copies of this module ''K''<sup>''n''</~~sup>.~~

	~~==Facts==~~
	~~Every [[simple module]] is indecomposable. The converse is not true in general, as is shown by the second example above.~~

	By looking at the [[endomorphism ring]] of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an [[idempotent element]] different from 0 and 1.<ref name="Jacobson" /~~> (If ''f'' is such an idempotent endomorphism of ''M'', then ''M'' is the direct sum of ker(''f'') and im(''f'').)~~

	A module of finite [[length of a module\|length]] is indecomposable if and only if its endomorphism ring is [[local ring\|local]]. Still more information about endomorphisms of finite-length indecomposables is provided by the [[Fitting lemma]].

	In the finite-length situation, decomposition into indecomposables is particularly useful, because of the [[Krull-Schmidt theorem]]: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic).~~<ref> Jacobson (2009), p~~. ~~115.<~~/~~ref>~~

	~~== Notes ==~~
	~~<references~~/>

	~~== References ==~~
	* {{Citation\| last=Jacobson\| first=Nathan\| author-link=Nathan Jacobson\| date=2009\| title=Basic algebra\| edition=2nd\| volume = 2 \| series= \| publisher=Dover\| isbn = 978-0-486-47187-7}}

	~~{{DEFAULTSORT:Indecomposable Module}}~~
	~~[[Category:Module theory]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7268946

2013-03-23T06:55:44Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7268946

← Older revision		Revision as of 08:55, 23 March 2013
Line 1:		Line 1:
	~~We can discover easy methods to accelerate computer by making the most from the built inside tools inside~~ a ~~Windows plus obtaining the Service Pack updates~~-~~speed up~~ a ~~PC and fix error~~. ~~Simply follow~~ a ~~limited protocols to immediately create your computer quick~~ than ~~ever.~~<br><br>~~Firstly~~, ~~you need to utilize your Antivirus or protection tool plus run a scan on the computer~~. It is ~~possible that the computer~~ is ~~afflicted with virus or malware~~ which ~~slows down your computer~~. ~~If there is nothing found in~~ the ~~scanning report~~, ~~it will~~ be ~~your RAM which cause the issue~~.~~<br><br>The 'registry'~~ is a ~~central database which stores information~~, ~~settings~~ and ~~choices for your computer~~. ~~It'~~s ~~really~~ the ~~most usual reason why XP runs slow plus should you fix this problem~~, ~~we can create a computer run a lot faster~~. ~~The problem is~~ the ~~fact that~~ the ~~'registry' shops~~ a ~~great deal of settings plus details about your PC~~.~~.. plus considering Windows must employ numerous of these settings~~, ~~any corrupted or damaged ones may directly affect~~ the ~~speed~~ of ~~your system.~~<br><br>~~Registry products have been designed~~ for ~~one purpose~~ - ~~to clean out the~~ '~~registry~~'. ~~This is the central database~~ that ~~Windows relies on to function. Without~~ this ~~database, Windows wouldn~~'~~t even exist. It~~'~~s thus important~~, ~~that~~ the ~~computer is regularly adding and updating the files inside it~~, ~~even when you~~'~~re browsing the Internet (like now). This~~ is ~~desirable, yet~~ the ~~difficulties happen when certain~~ of ~~those files become corrupt or lost~~. ~~This occurs a lot~~, ~~plus~~ it ~~takes~~ a ~~advantageous tool to fix it.~~<br><br>~~The final step is to create certain that we clean the registry of the computer~~. ~~The "registry"~~ is a ~~big database which shops important files, settings & choices, plus information~~. ~~Windows reads the files it requirements in order for it to run programs by this database. If the registry gets damaged~~, ~~afflicted~~, ~~or clogged up, then Windows may~~ not ~~be capable to correctly access~~ the ~~files it requires for it to load up programs~~. ~~As this happens~~, ~~problems and mistakes like~~ the ~~d3d9~~.~~dll error happen. To fix this and prevent future setbacks, you have to download and run~~ a ~~registry cleaning tool~~. ~~The highly suggested software~~ is ~~the "Frontline~~ [~~http://bestregistrycleanerfix.com/tune-~~up~~-utilities tuneup utilities~~]".<br><br>The ~~initial thing you need to do~~ is ~~to reinstall any program which shows~~ the ~~error~~. ~~It's typical for several computers to have specific programs which need this DLL to show~~ the ~~error when we try~~ and ~~load it up~~. If ~~you see a particular system show the error, you need to first uninstall which program~~, ~~restart your PC plus~~ then ~~resinstall~~ the ~~system again. This must substitute the damaged ac1st16.dll file~~ and ~~remedy the error~~.~~<br><br>The initial reason~~ a ~~computer will be slow~~ is ~~because it requirements~~ more ~~RAM~~. ~~You'll notice this problem right away~~, particularly ~~should~~ you have ~~less than~~ a ~~gig~~ of ~~RAM. Most modern computers come~~ with ~~a least~~ that ~~much~~. ~~While Microsoft states Windows XP can run on 128 MB~~, ~~it plus Vista want at least a gig to run smoothly and let you to run numerous programs at when~~. ~~Fortunately, the cost of RAM has dropped greatly, and you are able to get a gig installed for $100 or less~~.<br><br>Thus, the best thing to do whenever a computer runs slow is to buy an authentic and legal registry repair tool which would enable you eliminate all problems related to registry and aid we enjoy a smooth running computer.		{{Refimprove\|date=September 2011}}
			In [[abstract algebra]], a [[module (mathematics)\|module]] is '''indecomposable''' if it is non-zero and cannot be written as a [[direct sum of modules\|direct sum]] of two non-zero [[submodule]]s.<ref name="Jacobson"> Jacobson (2009), p. 111.</ref>

			Indecomposable is a weaker notion than [[simple module]]:
			simple means "no proper submodule" <math>N < M</math>,
			while indecomposable "not expressible as <math>N \oplus P = M</math>".

			A direct sum of indecomposables is called '''completely decomposable'''; this is weaker than being [[semisimple module\|semisimple]], which is a direct sum of [[simple module]]s.

			==Motivation==
			In many situations, all modules of interest are completely decomposable; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied. This is the case for modules over a
			[[field (mathematics)\|field]] or [[principal ideal domain\|PID]],
			and underlies [[Jordan normal form]] of [[linear operator\|operators]].

			==Examples==
			=== Field ===
			Modules over [[field (mathematics)\|field]]s are [[vector space]]s. A vector space is indecomposable if and only if its [[dimension (linear algebra)\|dimension]] is 1. So every vector space is completely decomposable (indeed, semisimple), with infinitely many summands if the dimension is infinite.<ref> Jacobson (2009), p. 111, in comments after Prop. 3.1.</ref>

			===PID===
			Finitely-generated modules over [[principal ideal domain\|PIDs]] are classified by the
			[[structure theorem for finitely generated modules over a principal ideal domain]]:
			the primary decomposition is a decomposition into indecomposable modules,
			so every finitely-generated module over a PID is completely decomposable.

			Explicitly, the modules of the form <math>R/p^n</math> for [[prime ideal]]s ''p'' (including ''p''=0, which yields ''R'') are indecomposable. Every finitely-generated ''R''-module is a direct sum of these. Note that this is simple if and only if ''n''=1 (or ''p''=0); for example, the cyclic group of order 4, '''Z'''/4, is indecomposable but not simple – it has the subgroup 2'''Z'''/4 of order 2, but this does not have a complement.

			Over the [[integer]]s '''Z''', modules are [[abelian group]]s. A finitely-generated abelian group is indecomposable if and only if it is [[isomorphic]] to '''Z''' or to a [[factor group]] of the form <math>\mathbf{Z}/p^n\mathbf{Z}</math> for some [[prime number]] ''p'' and some positive integer ''n''. Every [[finitely-generated abelian group]] is a direct sum of (finitely many) indecomposable abelian groups.

			There are, however, other indecomposable abelian groups which are not finitely generated; the [[rational number]]s '''Q''' form the simplest example.

			For a fixed positive integer ''n'', consider the ring ''R'' of ''n''-by-''n'' [[matrix (mathematics)\|matrices]] with entries from the [[real number]]s (or from any other field ''K''). Then ''K''<sup>''n''</sup> is a left ''R''-module (the scalar multiplication is [[matrix multiplication]]). This is [[up to isomorphism]] the only indecomposable module over ''R''. Every left ''R''-module is a direct sum of (finitely or infinitely many) copies of this module ''K''<sup>''n''</sup>.

			==Facts==
			Every [[simple module]] is indecomposable. The converse is not true in general, as is shown by the second example above.

			By looking at the [[endomorphism ring]] of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an [[idempotent element]] different from 0 and 1.<ref name="Jacobson" /> (If ''f'' is such an idempotent endomorphism of ''M'', then ''M'' is the direct sum of ker(''f'') and im(''f'').)

			A module of finite [[length of a module\|length]] is indecomposable if and only if its endomorphism ring is [[local ring\|local]]. Still more information about endomorphisms of finite-length indecomposables is provided by the [[Fitting lemma]].

			In the finite-length situation, decomposition into indecomposables is particularly useful, because of the [[Krull-Schmidt theorem]]: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic).<ref> Jacobson (2009), p. 115.</ref>

			== Notes ==
			<references/>

			== References ==
			* {{Citation\| last=Jacobson\| first=Nathan\| author-link=Nathan Jacobson\| date=2009\| title=Basic algebra\| edition=2nd\| volume = 2 \| series= \| publisher=Dover\| isbn = 978-0-486-47187-7}}

			{{DEFAULTSORT:Indecomposable Module}}
			[[Category:Module theory]]

123.24.235.76 at 15:54, 12 May 2012

2012-05-12T15:54:17Z

New page

We can discover easy methods to accelerate computer by making the most from the built inside tools inside a Windows plus obtaining the Service Pack updates-speed up a PC and fix error. Simply follow a limited protocols to immediately create your computer quick than ever. Firstly, you need to utilize your Antivirus or protection tool plus run a scan on the computer. It is possible that the computer is afflicted with virus or malware which slows down your computer. If there is nothing found in the scanning report, it will be your RAM which cause the issue. The 'registry' is a central database which stores information, settings and choices for your computer. It's really the most usual reason why XP runs slow plus should you fix this problem, we can create a computer run a lot faster. The problem is the fact that the 'registry' shops a great deal of settings plus details about your PC... plus considering Windows must employ numerous of these settings, any corrupted or damaged ones may directly affect the speed of your system. Registry products have been designed for one purpose - to clean out the 'registry'. This is the central database that Windows relies on to function. Without this database, Windows wouldn't even exist. It's thus important, that the computer is regularly adding and updating the files inside it, even when you're browsing the Internet (like now). This is desirable, yet the difficulties happen when certain of those files become corrupt or lost. This occurs a lot, plus it takes a advantageous tool to fix it. The final step is to create certain that we clean the registry of the computer. The "registry" is a big database which shops important files, settings & choices, plus information. Windows reads the files it requirements in order for it to run programs by this database. If the registry gets damaged, afflicted, or clogged up, then Windows may not be capable to correctly access the files it requires for it to load up programs. As this happens, problems and mistakes like the d3d9.dll error happen. To fix this and prevent future setbacks, you have to download and run a registry cleaning tool. The highly suggested software is the "Frontline [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]". The initial thing you need to do is to reinstall any program which shows the error. It's typical for several computers to have specific programs which need this DLL to show the error when we try and load it up. If you see a particular system show the error, you need to first uninstall which program, restart your PC plus then resinstall the system again. This must substitute the damaged ac1st16.dll file and remedy the error. The initial reason a computer will be slow is because it requirements more RAM. You'll notice this problem right away, particularly should you have less than a gig of RAM. Most modern computers come with a least that much. While Microsoft states Windows XP can run on 128 MB, it plus Vista want at least a gig to run smoothly and let you to run numerous programs at when. Fortunately, the cost of RAM has dropped greatly, and you are able to get a gig installed for $100 or less. Thus, the best thing to do whenever a computer runs slow is to buy an authentic and legal registry repair tool which would enable you eliminate all problems related to registry and aid we enjoy a smooth running computer.