Table Of ContentLattice-ordered Rings and Modules
Lattice-ordered Rings and
Modules
Stuart A. Steinberg
Toledo, OH, USA
Stuart A. Steinberg
Department of Mathematics
University of Toledo
Toledo, OH 43601
USA
[email protected]
ISBN 978-1-4419-1720-1 e-ISBN 978-1-4419-1721-8
DOI 10.1007/978-1-4419-1721-8
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009940319
Mathematics Subject Classification (2010): 06F25, 13J25, 16W60, 16W80, 06F15, 12J15, 13J05,
13J30, 12D15
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ToDiane
Stephen,David,andJulia
Preface
Alattice-orderedringisaringthatisalsoalatticeinwhicheachadditivetranslation
isorderpreservingandtheproductoftwopositiveelementsispositive.Manyring
constructionsproducearingthatcanbelattice-orderedinmorethanoneway.This
text is an account of the algebraic aspects of the theories of lattice-ordered rings
andofthoselattice-orderedmoduleswhichcanbeembeddedinaproductoftotally
ordered modules—the f-modules. It is written at a level which is suitable for a
second-year graduate student in mathematics, and it can serve either as a text for
acourseinlattice-orderedringsorasamonographforaresearcherwhowishesto
learnaboutthesubject;thereareover800exercisesofvariousdegreesofdifficulty
which appear at the ends of the sections. Included in the text is all of the relevant
background information that is needed in order to to make the theories that are
developedandtheresultsthatarepresentedcomprehensibletoreaderswithvarious
backgrounds.
In order to make this book as self-contained as possible it was necessary to in-
cludealargeamountofbackgroundmaterial.Thus,inthefirstchapterwehavecon-
structedtheDedekindandMacNeillecompletionsofapartiallyorderedset(poset)
anddevelopedenoughofuniversalalgebrasothatwecanpresentBirkhoff’schar-
acterizationofavarietyandsothatwecanalsoverifytheexistenceoffreeobjects
inavarietyofalgebras.Muchofthematerialonlattice-orderedgroups((cid:96)-groups)
inthesecondchapterappearsinthosebooksdevotedtothesubject.Whatisnewin
thisbookistheemphasison(cid:96)-groupswithoperators.Thisallowsforthecommon
developmentofbasicresultsabout(cid:96)-groups, f-rings,and f-modules.TheAmitsur-
Kuroshtheoryofradicalsisdevelopedfortheclassof(cid:96)-ringsinthesecondsection
of Chapter 2 . Still more background material is given in the first two sections of
Chapter 4 where the injective hull of a module, the Utumi maximal right quotient
ring,andtheringofquotientsandthemoduleofquotientswithrespecttoahered-
itarytorsiontheoryareconstructedandstudiedforaringwhichisnotnecessarily
unital. Also, the Artin–Schrier theory of totally ordered fields is given in the first
sectionofChapter5,andenoughofthetheoryofvaluationsonafieldispresented
inthesecondsectionsothatacompleteproofoftheHahnembeddingtheoremfor
awell-conditionedcommutativelattice-ordereddomaincanbegiven.
vii
viii Preface
Chapters3,4,5,and6constitutetheheartofthebook.Whilenoteveryknown
resultonthetopicsincludedispresented,enoughispresentedsoastomakethetext,
bywhichImeantheexercisesalso,reasonablycomplete.ThefirstsectionofChap-
ter3developsthebasictheoryof(cid:96)-ringsincludingthefactthatcanonicallyordered
matrixringshavenounital f-modules.Section4showsthatthefundamentalprocess
ofembeddingan f-algebrainaunital f-algebraismorecomplicatedthantheanal-
ogous embedding for algebras and cannot always be carried out. The fifth section
showshowtoconstructpowerseriestypeexamplesof(cid:96)-ringsand(cid:96)-modulesusing
a poset which is a partial semigroup and which is rooted in the sense that the set
ofupperboundsofeachelementisachain.Thebasicstructureof f-ringsisgiven
in the third section of Chapter 3 and some of the richer structure of archimedean
f-ringsisgiveninthesixthsection.Inthelasttwosectionsthestructureof(cid:96)-rings
in other varieties is examined. The seventh section studies those (cid:96)-rings that have
squarespositiveandgivesconditionsonapartiallyorderedgeneralizedsemigroup
forthelexicographicallyorderedsemigroupringtohavethisproperty.Thelastsec-
tionconsidersthose(cid:96)-ringswhichsatisfypolynomialconstraintsmoregeneralthan
thatofsquaresbeingpositive.Oneeffectoftheseconstraintsistocoalescetheset
ofnilpotentelementsintoasubringoranidealandtoforcean(cid:96)-semiprimeringto
lacknilpotentelements.Alsoincludedinthissectionisaproofofthecommutativity
ofanarchimedeanalmost f-ring.
Chapter 4 concentrates on the category of f-modules. The most conclusive re-
sultsoccur for asemiprime f-ring whose maximalright quotient ring isan f-ring
extensionandwhoseBooleanalgebraofannihilatorsisatomic.Inthethirdsection
necessary and sufficient conditions for the module (or ring) of quotients to be an
f-module(oran f-ring)extensionaregiven,andthestructureofrightself-injective
f-rings is given. The unique totally ordered right self-injective ring that does not
have an identity element is exhibited. The module and order theoretic properties
thatdeterminewhenanonsingular f-moduleisrelativelyinjectivearegiveninthe
fourthsection—therearenoinjectivesinthiscategoryof f-modules.Ausefulrep-
resentationofthefreenonsingular f-moduleisgiveninthelastsectionandthesize
ofadisjointsetinafree f-moduleisdetermined.
In a totally ordered field the set of values—those convex subgroups which are
maximal with respect to not containing a given nonzero element—becomes a to-
tallyorderedgroupundertheoperationinducedonitbymultiplicationinthefield.
AproofoftheHahnEmbeddingTheoremfortotallyorderedfieldsisgiveninthe
secondsectionofChapter5;namely,atotallyorderedfieldisembeddedinapower
seriesfieldwheretheexponentsbelongtothisvaluegroupofthefieldandtheco-
efficients are real numbers. Also, a totally ordered division ring is embedded in a
totallyordereddivisionalgebraoverthereals.InthethirdsectionofChapter5the
HahnEmbeddingTheoremisgivenforalattice-orderedcommutativedomainwhich
satisfiesafinitenesscondition,andanotherembeddingtheoremforasuitablycon-
ditioned(cid:96)-fieldintoaformalpowerseriescrossedproduct(cid:96)-ringisgiven.Also,the
theory of archimedean (cid:96)-fields is presented and lattice orders other than the usual
totalorderareconstructedforthefieldofrealnumbers.
