137.205.127.174: corrected typo in introduction

2012-12-11T16:04:18Z

corrected typo in introduction

← Older revision		Revision as of 18:04, 11 December 2012
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	~~I am Yasmin~~ and ~~was born~~ on ~~26 September 1984~~. ~~My hobbies~~ are ~~Herping~~ and ~~Bboying~~.<br><br>~~Look~~ into ~~my webpage~~; [http://~~www~~.~~mbtoutletuk~~.org.~~uk mbt outlet~~]		In the branch of [[abstract algebra]] called [[ring theory]], the '''double centralizer theorem''' can refer to any one of several similar results. These results concern the [[centralizer and normalizer\|centralizer of a subring]] ''S'' of a ring ''R'', denoted '''C'''<sub>''R''</sub>(''S'') in this article. It is always the case that '''C'''<sub>''R''</sub>('''C'''<sub>''R''</sub>(''S'')) contains ''S'', and a double centralizer theorem gives conditions on ''R'' and ''S'' that guarantee that '''C'''<sub>''R''</sub>('''C'''<sub>''R''</sub>(''S'')) is ''equal'' to ''S''.

			==Statements of the theorem==

			===Motivation===
			The centralizer of a subring ''S'' of ''R'' given by

			:<math>\mathrm{C}_R(S)=\{r\in R \mid rs=sr \text{ for all } s\in S\}.\,</math>

			Clearly '''C'''<sub>''R''</sub>('''C'''<sub>''R''</sub>(''S'')) &supe; ''S'', but it is not always the case that one can say the two sets are equal. The double centralizer theorems give conditions under which one can conclude that equality occurs.

			There is another special case of interest. Let ''M'' be a right ''R'' module and give ''M'' the natural left ''E''-module structure, where ''E'' is End(''M''), the ring of endomorphisms of the abelian group ''M''. Every map ''m''<sub>''r''</sub> given by ''m''<sub>''r''</sub>(''x'') = ''xr'' creates an additive endomorphism of ''M'', that is, an element of ''E''. The map ''r'' → ''m''<sub>''r''</sub> is a ring homomorphism of ''R'' into the ring ''E'', and we denote the image of ''R'' inside of ''E'' by ''R''<sub>''M''</sub>. It can be checked that the [[kernel (algebra)\|kernel]] of this canonical map is the [[annihilator (ring theory)\|annihilator]] Ann(''M''<sub>''R''</sub>). Therefore by an [[Isomorphism_theorem#First_isomorphism_theorem_2\|isomorphism theorem]] for rings, ''R''<sub>''M''</sub> is isomorphic to the quotient ring ''R''/Ann(''M''<sub>''R''</sub>). Clearly when ''M'' is a [[faithful module]], ''R'' and ''R''<sub>''M''</sub> are isomorphic rings.

			So now ''E'' is a ring with ''R''<sub>''M''</sub> as a subring, and '''C'''<sub>''E''</sub>(''R''<sub>''M''</sub>) may be formed. By definition one can check that '''C'''<sub>''E''</sub>(''R''<sub>''M''</sub>) = End(''M''<sub>''R''</sub>), the ring of ''R'' module endomorphisms of ''M''. Thus if it occurs that '''C'''<sub>''E''</sub>('''C'''<sub>''E''</sub>(''R''<sub>''M''</sub>)) = ''R''<sub>''M''</sub>, this is the same thing as saying '''C'''<sub>''E''</sub>(End(''M''<sub>''R''</sub>)) = ''R''<sub>''M''</sub>.

			===Central simple algebras===
			Perhaps the most common version is the version for [[central simple algebra]]s, as it appears in {{harv\|Knapp\|2007\|loc=p.115}}:

			'''Theorem''': If ''A'' is a finite dimensional central simple algebra over a field ''F'' and ''B'' is a simple subalgebra of ''A'', then '''C'''<sub>''A''</sub>('''C'''<sub>''A''</sub>(''B'')) = ''B'', and moreover the dimensions satisfy

			:<math>\mathrm{dim}_F(B)\cdot\mathrm{dim}_F(\mathrm{C}_A(B))=\mathrm{dim}_F(A).\,</math>

			===Artinian rings===
			The following generalized version for [[Artinian ring]]s (which include finite dimensional algebras) appears in {{harv\|Isaacs\|2009\|loc=p.187}}. Given a [[simple module\|simple]] ''R'' module ''U''<sub>''R''</sub>, we will borrow notation from the above motivation section including ''R''<sub>''U''</sub> and ''E''=End(''U''). Additionally, we will write ''D''=End(''U''<sub>''R''</sub>) for the subring of ''E'' consisting of ''R''-homomorphisms. By [[Schur's lemma]], ''D'' is a [[division ring]].

			'''Theorem''': Let ''R'' be a right Artinian ring with a simple right module ''U''<sub>''R''</sub>, and let ''R''<sub>''U''</sub>, ''D'' and ''E'' be given as in the previous paragraph. Then
			:<math>R_U=\mathrm{C}_E(\mathrm{C}_E(R_U))\,</math>.

			;Remarks:
			* In this version, the rings are chosen with the intent of proving the [[Jacobson density theorem]]. Notice that it only concludes that a particular subring has the centralizer property, in contrast to the central simple algebra version.
			* Since algebras are normally defined over commutative rings, and all the involved rings above may be noncommutative, it's clear that algebras are not necessarily involved.
			* If ''U'' is additionally a [[faithful module]], so that ''R'' is a right [[primitive ring]], then ''R''<sub>''U''</sub> is ring isomorphic to ''R''.

			===Polynomial identity rings===
			In {{harv\|Rowen\|1980\|loc=p.154}}, a version is given for [[polynomial identity ring]]s. The notation Z(''R'') will be used to denote the [[center of a ring]] ''R''.

			'''Theorem''': If ''R'' is a [[simple ring\|simple]] polynomial identity ring, and ''A'' is a simple Z(''R'') subalgebra of ''R'', then '''C'''<sub>''R''</sub>('''C'''<sub>''R''</sub>(''A'')) = ''A''.

			;Remarks
			* This version can be considered to be "between" the central simple algebra version and the Artinian ring version. This is because simple polynomial identity rings are Artinian,<ref>They are full matrix rings over polynomial identity division rings, according to {{harvtxt\|Rowen\|1980\|page=151}}</ref> but unlike the Artinian version, the conclusion still refers to all central simple subrings of ''R''.

			==Double centralizer property==
			{{main\|Balanced module}}

			A module ''M'' is said to have the ''[[balanced module\|double centralizer property]]'' or to be a ''[[balanced module]]'' if '''C'''<sub>''E''</sub>('''C'''<sub>''E''</sub>(''R''<sub>''M''</sub>)) = ''R''<sub>''M''</sub>, where ''E'' = End(''M'') and ''R''<sub>''M''</sub> are as given in the motivation section. In this terminology, the Artinian ring version of the double centralizer theorem states that simple right modules for right Artinian rings are balanced modules.

			== Notes ==
			<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
			{{Reflist}}

			==References==

			*{{citation \|last=Isaacs\|first= I. Martin \|title=Algebra: a graduate course \|series=Graduate Studies in Mathematics \|volume=100 \|note=Reprint of the 1994 original \|publisher=American Mathematical Society \|place=Providence, RI \|year=2009 \|pages=xii+516 \|isbn=978-0-8218-4799-2 \|mr=2472787}}

			*{{citation \|last=Knapp\|first=Anthony W. \|title=Advanced algebra \|series=Cornerstones \|publisher=Birkhäuser Boston Inc. \|place=Boston, MA \|year=2007 \|pages=xxiv+730 \|isbn=978-0-8176-4522-9 \|mr=2360434}}

			*{{citation \|last=Rowen\|first=Louis Halle \|title=Polynomial identities in ring theory \|series=Pure and Applied Mathematics \|volume=84 \|publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] \|place=New York \|year=1980 \|pages=xx+365 \|isbn=0-12-599850-3\|mr=576061}}

			<!--- Categories --->

			[[Category:Theorems in abstract algebra]]
			[[Category:Ring theory]]

en>TakuyaMurata at 10:47, 6 August 2012

2012-08-06T10:47:19Z

New page

I am Yasmin and was born on 26 September 1984. My hobbies are Herping and Bboying.<br><br>Look into my webpage; [http://www.mbtoutletuk.org.uk mbt outlet]

Locally profinite group - Revision history

137.205.127.174: corrected typo in introduction

en>TakuyaMurata at 10:47, 6 August 2012