Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions

372 Pages·2014·4.17 MB·English

by  Marco Fontana

Marco Fontana · Sophie Frisch
Sarah Glaz Editors
Commutative
Algebra
Recent Advances in Commutative
Rings, Integer-Valued Polynomials, and
Polynomial Functions
Commutative Algebra

Marco Fontana • Sophie Frisch (cid:129) Sarah Glaz
Editors
Commutative Algebra
Recent Advances in Commutative Rings,
Integer-Valued Polynomials, and Polynomial
Functions
123
Editors
MarcoFontana SophieFrisch
DipartimentodiMatematica MathematicsDepartment
UniversitàdegliStudiRomaTre GrazUniversityofTechnology
Roma,Italy Graz,Austria
SarahGlaz
DepartmentofMathematics
UniversityofConnecticut
Storrs,CT,USA
ISBN978-1-4939-0924-7 ISBN978-1-4939-0925-4(eBook)
DOI10.1007/978-1-4939-0925-4
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2014943239
Mathematics Subject Classification (2010): 13-06, 13Axx, 13Bxx, 13Cxx, 13Dxx, 13Exx, 13Fxx,
13Gxx,13Hxx,13Jxx
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Preface
Thisvolumeismainlytheoutcomeofaseriesofmini-coursesandaconferenceon
Commutativerings,integer-valuedpolynomialsandpolynomialfunctionsatTech-
nische Universität Graz, Austria, December 16–18 (mini-courses) and December
19–22(conference),2012.Italso containsa smallcollectionof invitedarticles by
some of the leading experts in the area, carefully selected for the impact of their
researchonthemajorthemesoftheconference.
Theaimofthismeetingwastopresentrecentprogressintheareaofcommutative
algebra, with primary emphasis on commutative ring theory and integer-valued
polynomialsalongwithconnectionstoalgebraicnumbertheory,algebraicgeometry
and homological algebra. The wide range of topics is reflected in the table of
contentsofthisvolume.Someoftheinvitedspeakerswhogavemini-courseshave
suppliedsurveysofthestateoftheartinnewlyemergingsubfields.
At the conference, we had the good fortune to see that our field attracts
excellentyoungmathematicians(whosubmittedgoodwork,bothindividuallyand
incollaborationwiththeoldguard)andthenotsogoodfortunetoseethatnoneof
theyoungresearchershavepermanentjobs.Maythefirsttrendremaininfullforce
andthesecondoneberemediedinthenearfuture!
Among the people and organizations who helped to make the conference and
this volume of proceedings possible, our special thanks go to the departmental
secretary Hermine Panzenböck and the doctoral student Roswitha Rissner, who,
between the two of them, shared all the hard work of organizing,from designing
the conference poster and implementing the website to applications for subsidies
andthepainstakingworkofbookkeepingandbalancingtheaccounts.Withouttheir
efforts,theconferencewouldnothavetakenplaceandthisvolumewouldnothave
seenthelightofday.
Wethankthesponsorsoftheconference:theprovinceofStyria,whosesubsidy
allowedustosponsorthetravelexpensesofsomegraduatestudentsandconference
participantsfromlow-incomecountries,andthefacultyofmathematicsandphysics
of Technische Universität Graz and the joint graduate school of natural sciences
“NAWIGraz”ofTechnischeUniversitätGrazandKarl-FranzensUniversitätGraz,
v
vi Preface
who together paid the travel expenses of all the invited speakers. Last, but not
least,wethanktheeditorialstaffofSpringer,inparticularElizabethLoew,fortheir
cooperation,hardworkandassistancewiththepresentvolume.
Rome,Italy MarcoFontana
Graz,Austria SophieFrisch
Storrs,Connecticut,USA SarahGlaz
December2013
Contents
WeakGlobalDimensionofPrüfer-LikeRings................................ 1
KhalidAdarbehandSalah-EddineKabbaj
Quasi-completeSemilocalRingsandModules ............................... 25
DanielD.Anderson
OntheTotalGraphofaRingandItsRelatedGraphs:ASurvey.......... 39
AymanBadawi
Prime Ideals in PolynomialandPower Series Rings over
NoetherianDomains............................................................. 55
ElaCelikbas,ChristinaEubanks-Turner,andSylviaWiegand
Integer-ValuedPolynomials:LookingforRegularBases(ASurvey)...... 83
Jean-LucChabert
OnBooleanSubringsofRings.................................................. 113
IvanChajdaandGüntherEigenthaler
OnaNewClassofIntegralDomainswiththePortableProperty.......... 119
DavidE.Dobbs,GabrielPicavet,andMartinePicavet-L’Hermitte
TheProbabilityThatInt (D)IsFree .......................................... 133
n
JesseElliott
Some Closure Operations in Zariski-Riemann Spaces
ofValuationDomains:ASurvey ............................................... 153
CarmeloAntonioFinocchiaro,MarcoFontana,andK.AlanLoper
TenProblemsonStabilityofDomains......................................... 175
StefaniaGabelli
TheDevelopmentofNon-NoetherianGradeandItsApplications ......... 195
LiviaHummel
vii
viii Contents
Stable Homotopy Theory, Formal Group Laws,
andInteger-ValuedPolynomials................................................ 213
KeithJohnson
HowtoConstructHugeChainsofPrimeIdealsinPowerSeriesRings ... 225
ByungGyunKangandPhanThanhToan
LocalizingGlobalPropertiestoIndividualMaximalIdeals ................ 239
ThomasG.Lucas
PrimeIdealsThatSatisfyHensel’sLemma................................... 255
StephenMcAdam
FinitelyStableRings............................................................. 269
BruceOlberding
IntegralClosureofRingsofInteger-ValuedPolynomialsonAlgebras .... 293
GiulioPeruginelliandNicholasJ.Werner
OnMonoidsandDomainsWhoseMonadicSubmonoidsAreKrull....... 307
AndreasReinhart
IntegralClosure.................................................................. 331
IrenaSwanson
OpenProblemsinCommutativeRingTheory................................ 353
Paul-JeanCahen,MarcoFontana,SophieFrisch,andSarahGlaz
Weak Global Dimension of Prüfer-Like Rings
KhalidAdarbehandSalah-EddineKabbaj
Abstract In 1969, Osofsky proved that a chained ring (i.e., local arithmetical
ring) with zero divisors has infinite weak global dimension; that is, the weak
globaldimensionofanarithmeticalringis 0,1,or1.In2007,BazzoniandGlaz
studied the homological aspects of Prüfer-like rings, with a focus on Gaussian
rings. They proved that Osofsky’s aforementioned result is valid in the context
of coherent Gaussian rings (and, more generally, in coherent Prüfer rings). They
closedtheirpaperwithaconjecturesustainingthat“theweakglobaldimensionof
aGaussianringis0,1,or1.”In2010,theauthorsofBakkarietal.(J.PureAppl.
Algebra214:53–60,2010)providedanexampleofaGaussianringwhichisneither
arithmeticalnor coherentand has an infinite weak global dimension.In 2011,the
authorsofAbuihlailetal.(J.PureAppl.Algebra215:2504–2511,2011)introduced
and investigated the new class of fqp-ringswhich stands strictly between the two
classes of arithmetical rings and Gaussian rings. Then, they proved the Bazzoni-
Glazconjectureforfqp-rings.Thispapersurveysafewrecentworksintheliterature
ontheweakglobaldimensionofPrüfer-likeringsmakingthistopicaccessibleand
appealing to a broad audience. As a prelude to this, the first section of this paper
providesfulldetailsforOsofsky’sproofofthe existenceofa modulewithinfinite
projectivedimensiononachainedring.Numerousexamples—arisingastrivialring
extensions—areprovidedtoillustratetheconceptsandresultsinvolvedinthispaper.
Keywords Weak global dimension (cid:129) Arithmetical ring (cid:129) fqp-ring (cid:129) Gaussian
ring (cid:129) Prüfer ring (cid:129) Semihereditary ring (cid:129) Quasi-projective module (cid:129) Trivial
extension
MathematicsSubjectClassification 13F05,13B05,13C13,16D40,16B50
K.Adarbeh(cid:129)S.-E.Kabbaj((cid:2))
DepartmentofMathematicsandStatistics,KFUPM,Dhahran31261,SaudiArabia
e-mail:[email protected];[email protected]
M.Fontanaetal.(eds.),CommutativeAlgebra:RecentAdvancesinCommutativeRings, 1
Integer-ValuedPolynomials,andPolynomialFunctions,DOI10.1007/978-1-4939-0925-4__1,
©SpringerScience+BusinessMediaNewYork2014