Extensions of Rings and Modules

442 Pages·2013·4.22 MB·English

by  Gary F. Birkenmeier

Gary F. Birkenmeier
Jae Keol Park
S. Tariq Rizvi
Extensions
of Rings
and Modules

Gary F. Birkenmeier (cid:2) Jae Keol Park (cid:2)
S. Tariq Rizvi
Extensions
of Rings
and Modules
GaryF.Birkenmeier S.TariqRizvi
DepartmentofMathematics DepartmentofMathematics
UniversityofLouisianaatLafayette TheOhioStateUniversityatLima
Lafayette,LA,USA Lima,OH,USA
JaeKeolPark
DepartmentofMathematics
BusanNationalUniversity
Busan,SouthKorea
ISBN978-0-387-92715-2 ISBN978-0-387-92716-9(eBook)
DOI10.1007/978-0-387-92716-9
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013942792
MathematicsSubjectClassification(2010): 16-02,16DXX,16E50,16E60,16GXX,16LXX,16N20,
16N60,16P60,16P70,16R20,16S34,16S35,16S36,16S50,16S60,16UXX,16W10,16W20,16W22,
46H10,46L05,46L08,46L35,46L40,46L45,15A12,15A21,15A33,65F35
©SpringerScience+BusinessMediaNewYork2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub-
lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany
errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect
tothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Dedicated to Our Parents, Families, and
Teachers
Preface
Since the discovery of the existence of the injective hull of an arbitrary module
independentlyin 1952 by Shoda and in 1953 by Eckmann and Schopf, there have
been numerous papers dedicated to the study and description of various types of
hullsor“minimal”extensionsofringsandmodulessatisfyingsomegeneralizations
of injectivity or of related conditions. The study of these overrings, overmodules
andextensionssatisfyingtheseconditionshasbeendealtwithindetailindifferent
papers. The question of when do certain properties transfer from any ring R to
its many types of extensions such as matrix ring extensions, (skew-)group rings,
polynomial ring extensions and Ore extensions, has also been of interest to many
for a long time. It appears that the research work on various types of hulls and
on the wide varieties of extensions is spread throughout the literature in disparate
researchpapers.Thus,abookwhichpresents(atleastsomepartof)thestateofthe
artonthesubjectandincludessomeofthemostrecentworkdoneonthesetopics,
isneeded.Thattherehasnotbeenacomprehensivetreatmentofthesetopics,isone
ofthemainreasonsforustowritethisresearchmonograph.
Sincethepropertiessuchhullsandextensionssatisfymaybeunlimitedandthus
cannot possibly be covered in one book, we wish to emphasize that in this mono-
graphwehavefocusedmainlyonhullsandextensionsthatsatisfycertainconditions
on direct summands. We have however also made an attempt to provide a general
theoryforhullsbelongingtoarbitraryclassesofrings(ormodules)whichcansat-
isfyotherproperties.
Amongotherreasons,theneedtopresenttheresultsonthetransferenceofcer-
tain algebraic properties to and from base rings and modules to various ring and
moduleextensionsbelongingtospecificclasses,inasystematicway,hasalsobeen
a motivation in writing this research monograph.To ensure some efficiency in the
transferofinformationbetweenaringoramoduletoitsoverringsorovermodules,
respectively, we use the notion of a “minimal essential extension” with respect to
belongingtoaspecialclass.Wetermthisa“hull”(belongingtothatparticularclass)
and show that the “closeness” of such hulls to the base ring (or module) enriches
thetransferofinformationfromthebasering(ormodule)tosuchahull.Thiswill
beshownasausefultoolinanalyzingthestructureofaring(orofamodule).Our
vii
viii Preface
desireistopresentresearchworkwehavebeeninvolvedinforovertwodecadesas
wellasthatdonebyothersonthevarioustopicsofthismonograph.Wealsowishto
showcasethevariousapplicationsofthisresearchtoAlgebraandFunctionalAnal-
ysis.
Ourviewinwritingthisbookisalsotostimulatenewandfurtherresearchonthe
topics presented. A number of open research problems are listed at the end of the
book to generate interest in research on these topics. It is our hope that the reader
willfindthematerialpresentedinanaccessibleandunifiedmanner.Thebookisin-
tendedforresearchmathematiciansinalgebraandanalysisandforadvancedgrad-
uatestudentsinmathematics.Eachsectionincludesexercisesofvaryingdegreesof
difficultyforgraduatestudents.
Whilewehaveattemptedtomakethismonographasself-containedaspossible,
ithasbeendifficultinviewofthelimitationsonthesizeofthisbook.Tokeepthe
book to a reasonable length, some proofs (including a few highly technical ones)
havebeenomittedandappropriatereferencestoresearchpapersorbookshavebeen
included. Some results have been included as exercises in various chapters, with
proper references, for a motivated reader. We have also listed other references re-
lated to the material presented in the book. These, and the brief historical notes
provided at the end of each chapter, should be useful for researchers interested in
furtherinvestigations.Therearemanyexcellentpaperswhichweregrettablycould
notincludeinthisbookinordertokeepitwithinamoderatelength.
WeareverythankfultoGangyongLee,CosminRoman,HenryE.Heatherly,and
TomaAlbu,fortheirnumerousconstructivecomments,painstakingproof-reading,
andsuggestionsforimprovementsinthisbook.ThetechnicalhelpprovidedbyCos-
minRomanandGangyongLee,andcorrectionsinseveralproofstheypointedout
havebeencrucialduringthepreparationofthismanuscript.Therearemanyothers
whohelpedinproof-readingvariouspartsofthebook.TheseincludeMohammad
Ashraf,XiaoxiangZhang,AsmaAli,ShakirAli,andFaizRizvi.Wearethankfulto
themfortheirtimeandefforts.Theerrorsthatstillmayremaininthebook,areour
ownfault.
There are others who have also played an important role, directly or indirectly,
intheformationofthisbook.Wearethankfulto(late)E.H.Feller,BrunoJ.Müller,
Joe W. Fisher, T.Y. Lam, Barbara L. Osofsky, S.K. Jain, Efraim P. Armendariz,
SergioLópez-Permouth,DinhVanHuynh,PereAra,EdmundR.Puczyłowski,and
MikhailChebotarfortheirinfluenceonourwork,wordsofencouragement,advice
andsupport.
TheworkonthebookwassupportedbygrantsfromTheOhioStateUniversity
at Lima, Mathematics Research Institute, The Ohio State University, Columbus,
Ohio, USA, Busan National University, Busan, South Korea, and the University
of Louisiana at Lafayette, Lafayette, Louisiana, USA. We express our gratitude to
theseinstitutionsfortheirsupport.S.T.Rizviwishestoacknowledgethesupportof
a 3-Year Stimulus Research Grant from the Mathematics Research Institute, Ohio
State University, Columbus, USA, for his work in the final stages of this mono-
graph.JaeKeolParkwassupportedbya2-YearResearchGrantofPusanNational
University.
Preface ix
This work would not have been possible without the help and support of our
families.Wecannoteverfullyexpressourheartfeltthankstoourrespectivewives,
Betti,Insook,andSeema,fortheirunstintingsupportthroughoutthepreparationof
thiswork.Theircontinuoussupport,andthatofourchildren,havebeenasourceof
strengthforusforwhichwearetrulyverygrateful.
We are grateful to Ann Kostant for the support and cooperation in the earlier
stages of processing of this research monograph. The helpful cooperation and pa-
tience we received from Allen Mann and Mitch Moulton in the final processing
of the book has been crucial. We greatly appreciate that and thank them for their
personal interest and professional support. We are also thankful to other staff at
Birkhäuserwhohavebeenhelpfulinourbookproject,includingTomGrasso.
Lafayette,Louisiana,USA GaryF.Birkenmeier
Busan,SouthKorea JaeKeolPark
Lima,Ohio,USA S.TariqRizvi
February12,2013
Introduction
AmongthemajoreffortsinRingTheory,onehasbeentofind,foragivenringR,a
“wellbehaved”overringS inthesensethatS hasbetterpropertiesthanR andsuch
thatsomeuseful informationcan transfer between R and S.Alternatively,givena
wellbehavedring,tofindconditionsdescribingthosesubringsforwhichthereisa
fruitfulinheritanceofpropertiesbetweenthegivenringanditssubrings.Asimilar
quest between a module and an overmodule has been pursued in Module Theory.
These have been important topics of research and have been crucial in the devel-
opment of Algebra—especially of Ring and Module Theory. Having yielded such
important classes of rings and modules, as the rings and modules of quotients, in-
jectivehullsandrightorders,thisquesthasbeentrulyrewardingtoringandmodule
theorists.
Anotherefforthasbeen,toinvestigatewhenpropertiesofagivenring R trans-
fertoitsvariousringextensionsandviceversa.Anumberofresearchpapershave
beenpublishedoninvestigationstoaddresssuchquestions.Theringextensions(for
example,polynomialextensions,matrixringextensions,triangularmatrixringex-
tensions,groupringextensions,andskewgroupringextensions,etc.)formimpor-
tantclassesofringsandhavebeenafocusofextensiveresearch.Whileresultson
someparticulartypesofringextensionsandthetransferofsomelimitedalgebraic
propertieshavebeenincludedinafewexistingresearchbooksorgraduatetexts,it
appearsthatthereispresentlynoresearchmonographcoveringthewidevarietiesof
extensionsandmuchoftherecentworkisspreadindisparateresearchpapers.
Thefocusofourbookisrelatedtothetwoquestsmentionedabove.Aswemen-
tioned in the preface, for a given ring R (or a given module M), we consider a
“minimalessentialextension”ofR (orofM)withrespecttobelongingtoapartic-
ularclass.Wecallthisa“hull”ofR(orofM)belongingtothatparticularclassand
showthatsuchhullslieclosertotheringR (ortothemoduleM)thanitsinjective
hull.ThisinturnallowsforabettertransferofinformationbetweenR (orM)and
the hull of R (or of M) from these classes than between R (or M) and its injec-
tivehull.ThesehullsprovetobeusefultoolsforthestudyofthestructureofR (or
ofM).
xi
xii Introduction
Whilesomeofthetechniquespresentedherecanbeappliedinmoregeneralset-
tings, our focus in this book is on certain properties of rings and modules related
totheirdirectsummandsanddirectsums.In1940,R.Baer[35]introducedtheno-
tion of an injective module and showed that a module M is injective if and only
R
if, whenever M ≤N , M is a direct summand of N . This generalization of a
R R R R
vector space is one of the cornerstones of Module Theory. The notion of injectiv-
ityanditsgeneralizationshavebeenadirectionofextensiveresearch.Theneedto
studygeneralizationsofinjectivityarisesfromthefactthatsuchclassesofmodules
properlycontaintheclassofinjectivemoduleswhilestillenjoyingsomeworthwhile
advantagesof injectivemodules.One such generalizationthat has been of interest
foraboutthreedecades,isthenotionofextending(orCS)modules,namelymodules
inwhicheverysubmoduleisessentialinadirectsummand.Thisnotionwasexplic-
itly named for the case of rings (as CS-rings) by Chatters and Hajarnavis in 1977
[119]andwasalsostudiedearlierbyUtumi[398].Itiseasytoseethatsuchamod-
uleisacommongeneralizationofinjectiveandsemisimplemodules.Analogously,
amoduleM iscalledanFI-extendingmoduleifeveryfullyinvariantsubmoduleof
M isessentialinadirectsummandofM.Theseclassesofmodulesandrelatedno-
tionswillformanimportantfocusofresultsinthisbookbecauseoftheinteresting
connections,aswewillseelater,toothertopicsofourstudy.
AmongoverringsofR,itsrightringsofquotientsprovidehandytoolsforstudy-
ingR.HowevertheybecomeuselesswhenR coincideswithitsmaximalrightring
ofquotientssuchaswhenR isarightKaschring.Tostudyoverringsofsuchrings
one can consider classes of rings that lie between R and its right injective hull
E(R ). This motivates the notion of essential overring extensions that we present
R
inChap.7andfurtherutilizeinlaterchapters.WecallanoverringS ofaringR,a
rightessentialoverringof R if R isessentialin S .Thisnotionwillprovetobe
R R
ausefultool.ThestudyofsuchextensionsisalsomotivatedbyaresultinChap.8
which shows that any right essential overring of a right FI-extending ring is right
FI-extending.Therefore,allrightessentialoverringsofarightFI-extendinghull(if
it exists) of a ring R are right FI-extending. A ring is called quasi-Baer if the left
annihilator of an ideal is generated by an idempotent. Any right and left essential
overringofaquasi-Baerringisquasi-Baer.Alsorightessentialoverringsprovidea
naturalsettingfordefiningthenotionofringhullsinChap.8.
In1936,MurrayandvonNeumann[311]developedthetheoryofvonNeumann
∗
algebras(alsocalledW -algebras)inanattempttoprovidearigorousmathematical
modelforquantumtheory(seealso[403–406],and[407]).Theirtheorywasbased
∗
onringsofoperatorsonaHilbertspace.Rickart[353]in1946studiedC -algebras
(i.e., Banach ∗-algebras such that (cid:4)xx∗(cid:4)=(cid:4)x(cid:4)2) which satisfy the condition that
therightannihilatorofeverysingleelementisgeneratedbyaprojection(anidem-
potenteiscalledaprojectionife=e∗).RickartalsoshowedthatallvonNeumann
algebrassatisfythisproperty(i.e.,therightannihilatorofanyelementisgenerated
∗
byaprojection).ThesealgebraswerelaternamedRickartC -algebrasbyKaplan-
sky.
MotivatedbytheworkofMurray,vonNeumann,andRickart,Kaplanskyinthe
1950s showed that von Neumann algebras, in fact, satisfied a stronger annihilator