Advances in Commutative Algebra: Dedicated to David F. Anderson

280 Pages·2019·4.297 MB·English

by  Ayman Badawi, Jim Coykendall

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
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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
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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.