Cyclic Modules and the Structure of Rings (Oxford Mathematical Monographs)

177 Pages·2012·1.432 MB·English

by  S.K. Jain, Ashish K. Srivastava, Askar A. Tuganbaev

AskarA.Tuganbaev
ArithmeticalRingsandEndomorphisms
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Askar A. Tuganbaev
Arithmetical Rings
and
Endomorphisms
|
MathematicsSubjectClassification2010
16D25,16D40,13F05,16D50,16D70,16E60,16P40,16P50,16P60,13F05
Author
Prof.Dr.AskarA.Tuganbaev
NationalResearchUniversityMPEI
DepartmentofHigherMathematics
Krasnokazarmennaya14
Moscow111250
RussianFederation
[email protected]
ISBN978-3-11-065889-7
e-ISBN(PDF)978-3-11-065982-5
e-ISBN(EPUB)978-3-11-065915-3
LibraryofCongressControlNumber:2019939556
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TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
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Preface
This monograph isa comprehensive account of not necessarily commutative arith-
meticalrings,examiningstructuralandhomologicalpropertiesofmodulesoverarith-
metical rings and summarizing the interplay between arithmetical rings and other
rings.Moduleswithextensionpropertiesofsubmoduleendomorphismsarealsosys-
tematicallystudiedinthisbook.
Graduatestudentsandresearchersinterestedinringtheoryandmoduletheorywill
find this book particularly valuable. Containing numerous examples, Arithmetical
RingsandEndomorphismsisalargelyself-containedandaccessibleintroductionto
thetopic,assumingasolidunderstandingofbasicalgebra.
ThestudyissupportedbytheRussianScienceFoundation,projectno.16-11-10013.
Keywords:arithmeticalring,distributivemodule,flatmodule,localizationbyamax-
imalideal,Bezoutring,Hermitering,endomorphism-extendablemodule,automor-
phism-extendablemodule,automorphism-invariantmodule,injectivemodule,quasi-
injectivemodule,stronglysemiprimering
https://doi.org/10.1515/9783110659825-201

Introduction
This book consists of two parts. In Part I, “Arithmetical rings,” we systematically
studynotnecessarilycommutativeringswithdistributivelatticeoftwo-sidedideals.
In Part II, “Extension of automorphisms and endomorphisms,” we study modules
with the extension property of automorphisms and endomorphisms from submod-
ulestothewholemodule,andalsocharacteristicsubmodulesoftheirinjectivehulls.
Themaincontentofthebook
ThemainresultsofSection1“SaturatedIdealsandLocalizations”areTheorems1A,
1Band1C.
1ATheorem(Tuganbaev[172]). ArightinvariantringA isarithmeticalifandonlyif
foritsmaximalidealM,all{A\M}-saturatedidealsoftheringAformachainwith
respecttoinclusion.
1BTheorem(Jensen[98]). AcommutativeringAisarithmeticalifandonlyifforits
maximalidealM,thelocalizationA isauniserialring.
M
1CTheorem(Tuganbaev[157]). ArightinvariantringAisanarithmeticalsemiprime
ringifandonlyifforitsmaximalidealM,therightlocalizationA existsandisaright
M
uniserialdomain.
ThemainresultsofSection2,“FinitelyGeneratedModulesandDiagonalizability,”are
Theorems2Aand2B.
2ATheorem(Golod[77]). IfAisacommutativering,thenAisarithmeticalifandonly
ifB+r(X) = r(X/XB)foreveryfinitelygeneratedA-moduleXandeachidealBofthe
ringA.
2BTheorem(Tuganbaev[187]). IfAisarightinvariant,diagonalizable¹ring,thenB+
r(X) = r(X/XB)foreveryfinitelygeneratedrightA-moduleXandeachidealBofthe
ringA.
ThemainresultsofSection3“Ringswithflatandquasiprojectiveideals”arethefol-
lowingTheorems3A,3Band3C.
3ATheorem(Tuganbaev[157,161,182]). ForaninvariantsemiprimeringA,thefol-
lowingconditionsareequivalent.
1 Thedefinitionofadiagonalizableringisgivenin2.2.1.
https://doi.org/10.1515/9783110659825-202
VIII | Introduction
1)Aisanarithmeticalring.
2)EverysubmoduleofanyflatA-moduleisaflatmodule.
3)EveryfinitelygeneratedidealoftheringAisaquasiprojectiverightA-module.
3BTheorem(Jensen[98]). AcommutativeringAisanarithmeticalsemiprimeringif
andonlyifeverysubmoduleofanyflatA-moduleisaflatmodule.
3CTheorem(Tuganbaev[182]). IfAisaninvariantring,thenAisanarithmeticalring
ifandonlyifeveryoneofitsfinitelygeneratedidealsisaquasiprojectiverightA-mod-
ulesuchthatallendomorphismscanbeextendedtoendomorphismsofthemodule
A .
A
ThemainresultsofSection4“HermiteringsandPiercestalks”areTheorems4Aand
4B.
4ATheorem(Tuganbaev[183]). IfAisarightPPBezoutringwithoutnoncentralidem-
potents,thenAisaHermitering.
4BTheorem(Tuganbaev[183]). IfAisaBezoutringsuchthateveryPiercestalkisa
serialring,thenAisadiagonalizablering.
ThemainresultsofSection5“BezoutRings,Krulldimension”areTheorems5A,5B
and5C.
5A Theorem (Tuganbaev [183]). If A is a Bezout exchange ring without noncentral
idempotents,thenAisadiagonalizablering.
5BTheorem(Tuganbaev[187]). IfAisarightinvariant,arightBezout,exchangering,
thenB+r(X)= r(X/XB)foreveryfinitelygeneratedrightA-moduleXandeachideal
BoftheringA.
5CTheorem(Tuganbaev[188]). If A isacommutativearithmeticalring,then A has
theKrulldimensionifandonlyifeveryfactorringoftheringAisfinite-dimensional
anddoesnothaveidempotentproperessentialideals.
ThemainresultsofSection6“Semi-ArtinianandNonsingularModules”areTheorems
6A,6Band6C.
6ATheorem(Tuganbaev[184]). IfM isasemi-Artinian²module,thenM isanauto-
morphism-extendablemoduleifandonlyifMisanautomorphism-invariantmodule.
6BTheorem(Tuganbaev[174]). IfMisamoduleoveranArtinianserialring,thenM
isanautomorphism-extendablemoduleifandonlyifMisaquasi-injectivemodule.
2 AmoduleMissaidtobesemi-Artinianifeachofitsnonzerofactormoduleshasasimplesubmodule.
Introduction | IX
6C Theorem(Tuganbaev [176]). Let M = T ⊕ U, where T isaninjectivemodule, U
isanonsingular module,andHom(T󸀠,U) = 0foranysubmodule T󸀠 ofthemodule
T.ThemoduleMisautomorphism-extendableifandonlyifU isanautomorphism-
extendablemodule.
ThemainresultsofSection7“ModulesoverStronglyPrimeandStronglySemiprime
Rings”areTheorems7Aand7B.
7ATheorem(Tuganbaev[179]). IfAisarightstronglyprimering,thenarightA-mod-
uleMisautomorphism-invariantifandonlyifeitherMisasingularautomorphism-
invariantmoduleorMisaninjectivemodule.
7B Theorem(Tuganbaev [176]). If M isa right module over an invarianthereditary
domainA,thenthefollowingconditionsareequivalent.
1)Misanautomorphism-extendable(stronglyautomorphism-extendable)module.
2)Misanendomorphism-extendable(stronglyendomorphism-extendable)module.
3)EitherMisaquasi-injectivesingularmoduleorMisaninjectivemodulethatisnot
singular,orM = X⊕Y,whereXisaninjectivesingularmoduleandthemoduleY is
isomorphictoanonzerosubmoduleinQ ,whereQisadivisionringoffractionsof
A
thedomainA.
The mainresults of Section 8 “Endomorphism-extendable Modules and Rings” are
Theorems8Aand8B.
8ATheorem(Tuganbaev[167]). AringA isarightendomorphism-extendable,right
nonsingularringifandonlyifA=B×C,whereBisarightinjectiveregularring,Cisa
leftinvariant,reducedBaerringandCisarightcompletelyintegrallyclosedsubring
ofitsmaximalrightringsoffractionsQ.
8BTheorem(Tuganbaev[162]). AringAisaright(left)Noetherianringsuchthatall
cyclicright(left)modulesareendomorphism-extendableifandonlyifA =A ×⋅⋅⋅×A ,
1 n
whereA iseitherasimpleArtinianringorauniserialArtinianring,oraninvariant
i
hereditaryNoetheriandomain,i=1,...,n.
ThemainresultsofSection9“Automorphism-invariantnonsingularmodulesandthe
rings”areTheorems9A,9B,9C,9D.
9ATheorem(Tuganbaev[185]). LetAbearightstronglysemiprimering.IfXisaright
A-moduleandthereexistsanessentialrightidealBoftheringAsuchthatXisinjec-
tivewithrespecttothemoduleB ,thenXisaninjectivemodule.
A
9BTheorem(Tuganbaev[189]). IfAisaringwithrightGoldieradicalG,thenthefol-
lowingconditionsareequivalent.