Advances in Commutative Algebra: Dedicated to David F. Anderson PDF
Preview Advances in Commutative Algebra: Dedicated to David F. Anderson
Trends in Mathematics Ayman Badawi Jim Coykendall Editors Advances in Commutative Algebra Dedicated to David F. Anderson Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising fromconferences andlecture series focusing onaparticulartopicfrom anyarea of mathematics.Itsaimistomakecurrentdevelopmentsavailabletothecommunityas rapidlyaspossiblewithoutcompromisetoqualityandtoarchivetheseforreference. More information about this series at http://www.springer.com/series/4961 Ayman Badawi Jim Coykendall (cid:129) Editors Advances in Commutative Algebra Dedicated to David F. Anderson Editors Ayman Badawi JimCoykendall Department ofMathematics andStatistics Department ofMathematical Sciences American University of Sharjah Clemson University Sharjah, UnitedArab Emirates Clemson, SC,USA ISSN 2297-0215 ISSN 2297-024X (electronic) Trends inMathematics ISBN978-981-13-7027-4 ISBN978-981-13-7028-1 (eBook) https://doi.org/10.1007/978-981-13-7028-1 LibraryofCongressControlNumber:2019933704 ©SpringerNatureSingaporePteLtd.2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface When David F. Anderson retired in September 2017, his many friends and col- leagues from around theworld felt it appropriate to acknowledge his contributions to the wide-ranging area of commutative algebra by writing an edited book in his honor.Theenthusiasticresponsetothisprojectleddirectlytothiseditedbook;we are most thankful to all the contributors who helped to make this work possible. However, the material appearing in this book is not of the usual conference pro- ceedings type: The editors have tried to present a balanced mix of survey papers, which will enable experts and non-experts alike to get a good overview of devel- opments across a range of areas of commutative algebra outlining the work of DavidF.Anderson,alongwithresearchpaperspresentingsomeofthemostrecent developments in commutative algebra. Every effort has been made to make these research papers easily accessible in their introductory sections. We hope that the material will be of interest to both beginning graduate students and experienced researchersalike.Thetopicscoveredare,inevitably,justacrosssectionofthevast areas of commutative algebra, but they reflect in a strong way the areas in which David F. Anderson contributed so much. Thebookcontainssurveysandrecentresearchdevelopments.Finally,wewould like to express our sincere thanks to the colleagues who contributed papers so enthusiastically,tothemanyexpertswhoactedasrefereesforallthepapers,tothe professional staff at Springer, and in particular to Shamim Ahmad and Shubham Dixit, for their help in producing a volume which we hope is an appropriate recognition of our friend David F. Anderson. Sharjah, United Arab Emirates Ayman Badawi Clemenson, USA Jim Coykendall v Contents David Anderson and His Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 1 D. D. Anderson On F-Semi-homogeneous Integral Domains . . . . . . . . . . . . . . . . . . . . . 7 D. D. Anderson and Muhammad Zafrullah t-Local Domains and Valuation Domains . . . . . . . . . . . . . . . . . . . . . . . 33 Marco Fontana and Muhammad Zafrullah Strongly Divided Pairs of Integral Domains . . . . . . . . . . . . . . . . . . . . . 63 Ahmed Ayache and David E. Dobbs Finite Intersections of Prüfer Overrings . . . . . . . . . . . . . . . . . . . . . . . . 93 Bruce Olberding Strongly Additively Regular Rings and Graphs . . . . . . . . . . . . . . . . . . 113 Thomas G. Lucas On t-Reduction and t-Integral Closure of Ideals in Integral Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Salah Kabbaj Local Types of Classical Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 L. Klingler and W. Wm. McGovern pffiffiffiffiffiffiffi How Do Elements Really Factor in ZZ[ (cid:1)5]? . . . . . . . . . . . . . . . . . . . . 171 Scott T. Chapman, Felix Gotti and Marly Gotti David Anderson’s Work on Graded Integral Domains . . . . . . . . . . . . . 197 Gyu Whan Chang and Hwankoo Kim Divisor Graphs of a Commutative Ring . . . . . . . . . . . . . . . . . . . . . . . . 217 John D. LaGrange Isomorphisms and Planarity of Zero-Divisor Graphs . . . . . . . . . . . . . . 245 Jesse Gerald Smith Jr. vii About the Editors Ayman Badawi is Professor at the Department of Mathematics and Statistics, the AmericanUniversityofSharjah,theUnitedArabEmirates.HeearnedhisPh.D.in AlgebrafromtheUniversityofNorthTexas,USA,in1993.Heisanactivemember of the American Mathematical Society and honorary member of the Middle East CenterofAlgebraanditsApplications.Hisresearchinterestsincludecommutative algebra, pi-regular rings, and graphs associated to rings. Jim Coykendall is Professor of Mathematical Sciences at Clemson University, South Carolina, USA. He earned his Ph.D. from Cornell University in 1995, and has held various academic positions at the California Institute of Technology, the UniversityofTennessee,CornellUniversity,LehighUniversity,andNorthDakota State University. He has successfully guided 12 Ph.D. students. His research interests include commutative algebra and number theory. ix Introduction DavidF.Andersonisagreatteacher,afriend,anda supportivecolleague.Ihaveadmiredhishugeknowledgeand hispassionforcommutativeringtheory.Itisaprivilegethat Ihadtheopportunitytowrite17paperswithhimsofar.Inspite ofhisbusyschedule,hewasalwayswillingtofindthetimefor metodiscussmathematicsandresearchprojects. —AymanBadawi IfirstmetDavidinthesummerof1987whenIwasfortunate enoughtohavebeenassignedtohimashisresearchmenteeat theUniversityofTennessee’sREUprogram,andIwaslucky enough to be able to repeat the experience in the summer of 1988. These two summers were incredibly formative in my personal mathematical experience. During those summers, I discovered David to be incredibly gifted at sharing his understandingofmathematics(heisprobablythebestexpositor of mathematics that I know). It is interesting to note that althoughmyPhDwastechnicallyinalgebraicnumbertheory atCornellUniversity,myworknowmorecloselyapproximates themathematicsthatDavidturnedmeontointhelate80s.The “AndersonEffect”hastrulyshapedmycareerbothbeforeand after my graduate school education. I have greatly admired David’sworkfrombothnearandafar.Ihavebeenprivilegedto havebenefittedfromhismathematics,andIcontinuetoaspire tohisaccomplishmentswithmathematicsanditsexpositionand teaching. —JimCoykendall xi xii Introduction DavidF.Anderson DavidF.AndersonwhoretiredinSeptember2017isoneoftheleadingalgebraists ofhisgeneration.HewasborninFortDodge,Iowa,andgrewupinGowrie,Iowa, atownofabout1000peoplelocated80milesnorthwestofDesMoines.Hisfather was a rural mail carrier, and his mother was an elementary school teacher. His father’s cousin was E. F. Lindquist, a professor at the University of Iowa, who helped develop the ACT test and the GED examination and has the patent for the first optical-mark scanner. Much of his childhood was spent riding bikes, playing football, delivering newspapers, launching rockets, and being a Boy Scout. His interest inAbstract Algebrastartedfrom asummerprogramthatheattended atthe UniversityofIowain1966.HegraduatedfromPrairieCommunityHighSchoolin 1967alongwith58others,includinganotherfuturemathematician(histwinbrother Dan Anderson, now at the University of Iowa). DavidF.AndersonattendedIowaStateUniversity,wherehereceivedhisBSand MS in 1971. His mathematical skills were put to good use “counting cars” for the IowaStateHighwayCommissionduringthesummersof1967and1968.Duringthe summerof1969,heattendedanREUprograminmathematicsatSt.OlafCollege. Then, he went to the University of Chicago on an NSF Fellowship, where he receivedhisPh.D.in1976.HejoinedtheUniversityofTennesseein1976andhad been Associate Department Head for Graduate Studies since 2001. He had been activeinourREUprogramandwasanAPCalculusGraderandTableLeaderfrom 1987 to 1998. He is married with two grown children and five grandchildren. He enjoysreadingabouthistoryandreligionandinparticularenjoyslisteningtocollege courseson CDs while commuting to work each day. David F. Anderson published more than 160 papers in different branches of commutativealgebra.HiscompleterecordcanbeseenonMathScienceNet.Many of his publications appeared in very prestigious journals (e.g., Proceedings of the American Mathematical Society, Journal of Algebra, Journal of Pure and Applied Algebra,CommunicationsinAlgebra,JournalofAlgebraandItsApplications).He was a keynote speaker for several American Mathematical Society meetings.
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