Algebra and Applications Piotr Krylov Askar Tuganbaev Formal Matrices Formal Matrices Algebra and Applications Volume 23 Series editors: Michel Broué Université Paris Diderot, Paris, France Alice Fialowski Eötvös Loránd University, Budapest, Hungary Eric Friedlander University of Southern California, Los Angeles, USA Iain Gordon University of Edinburgh, Edinburgh, UK John Greenlees Sheffield University, Sheffield, UK Gerhard Hiß Aachen University, Aachen, Germany Ieke Moerdijk Radboud University Nijmegen, Nijmegen, The Netherlands Christoph Schweigert Hamburg University, Hamburg, Germany Mina Teicher Bar-Ilan University, Ramat-Gan, Israel Alain Verschoren University of Antwerp, Antwerp, Belgium Algebra and Applications aims to publish well written and carefully refereed monographs with up-to-date information about progress in all fields of algebra, its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and(co)homology theory, physicsanddiscrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C*-algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods incomputer science. More information about this series at http://www.springer.com/series/6253 Piotr Krylov Askar Tuganbaev (cid:129) Formal Matrices 123 Piotr Krylov Askar Tuganbaev TomskState University National Research University MPEI Tomsk Moscow Russia Russia Thestudy is supported bytheRussian Science Foundation(project no.*16-11-10013) ISSN 1572-5553 ISSN 2192-2950 (electronic) Algebra andApplications ISBN978-3-319-53906-5 ISBN978-3-319-53907-2 (eBook) DOI 10.1007/978-3-319-53907-2 LibraryofCongressControlNumber:2017932631 MathematicsSubjectClassification(2010): 16D20,16D40,16D50,16D90,16E20,16E60,15A15 ©SpringerInternationalPublishingAG2017 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. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents 1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1 2 Formal Matrix Rings .... ..... .... .... .... .... .... ..... .... 3 2.1 Construction of Formal Matrix Rings of Order 2. .... ..... .... 3 2.2 Examples of Formal Matrix Rings of Order 2 ... .... ..... .... 8 2.3 Formal Matrix Rings of Order n(cid:1)2... .... .... .... ..... .... 10 2.4 Some Ideals of Formal Matrix Rings .. .... .... .... ..... .... 14 2.5 Ring Properties.. .... ..... .... .... .... .... .... ..... .... 18 2.6 Additive Problems ... ..... .... .... .... .... .... ..... .... 24 3 Modules over Formal Matrix Rings.. .... .... .... .... ..... .... 31 3.1 Initial Properties of Modules over Formal Matrix Rings..... .... 31 3.2 Small and Essential Submodules . .... .... .... .... ..... .... 44 3.3 The Socle and the Radical .. .... .... .... .... .... ..... .... 48 3.4 Injective Modules and Injective Hulls.. .... .... .... ..... .... 51 3.5 Maximal Rings of Fractions. .... .... .... .... .... ..... .... 57 3.6 Flat Modules ... .... ..... .... .... .... .... .... ..... .... 64 3.7 Projective and Hereditary Modules and Rings ... .... ..... .... 69 3.8 Equivalences Between the Categories R-Mod, S-Mod, and K-Mod. .... .... ..... .... .... .... .... .... ..... .... 76 3.9 Hereditary Endomorphism Rings of Abelian Groups .. ..... .... 85 4 Formal Matrix Rings over a Given Ring.. .... .... .... ..... .... 89 4.1 Formal Matrix Rings over a Ring R... .... .... .... ..... .... 89 4.2 Some Properties of Formal Matrix Rings over R . .... ..... .... 97 4.3 Characterization of Multiplier Matrices. .... .... .... ..... .... 100 4.4 Classification of Formal Matrix Rings . .... .... .... ..... .... 104 4.5 The Isomorphism Problem .. .... .... .... .... .... ..... .... 111 4.6 Determinants of Formal Matrices. .... .... .... .... ..... .... 115 4.7 Some Theorems About Formal Matrices.... .... .... ..... .... 122 v vi Contents 5 Grothendieck and Whitehead Groups of Formal Matrix Rings. ........ 129 5.1 Equivalence of Two Categories of Projective Modules. ..... .... 129 5.2 The Group K ðA;BÞ.. ..... .... .... .... .... .... ..... .... 133 0 5.3 K Groups of Formal Matrix Rings ... .... .... .... ..... .... 138 0 5.4 The K Group of a Formal Matrix Ring.... .... .... ..... .... 142 1 5.5 K and K Groups of Matrix Rings of Order n(cid:1)2 ... ..... .... 146 0 1 References.... .... .... .... ..... .... .... .... .... .... ..... .... 151 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 155 Symbols E Matrix unit ij jAj Determinant of a matrix A dðAÞ Determinant of a formal matrix A A(cid:3)B Kronecker product of matrices A and B Mðn;RÞ Ring of all n(cid:4)n matrices over the ring R Mðn;R;fs gÞorMðn;R;RÞ Ring of all formal matrices oforder n over thering ijk R with multiplier system fs g¼R ijk Mðn;R;sÞ Ring of all formal matrices oforder n over thering R with multiplier s R(cid:5) Opposite ring to a ring R PðRÞ Prime radical of the ring R JðRÞ Jacobson radical of the ring R CðRÞ Center of the ring R QðRÞ Maximal left ring offractions of the ring R R(cid:4)S Direct product of rings R and S A (cid:6)...(cid:6)A Direct sum of modules A ;...;A 1 n 1 n Ker u or KerðuÞ Kernel of the homomorphism u Imu or ImðuÞ Image of the homomorphism u Rad A Radical of the module A Soc A Socle of the module A Z Closure of the submodule Z of the module A A^ Injective hull of the module A A(cid:7) Character module of the module A limA Direct limit of modules A i i !I R-Mod (Mod-R) Category of left (right) modules over the ring R PðRÞ CategoryoffinitelygeneratedprojectiveR-modules PðAÞ Category offinitely A-projective modules End A or End ðAÞ Endomorphism ring of an R-module A R R Biend A Biendomorphism ring of an R-module A R EndG Endomorphism ring of an Abelian group G vii viii Symbols Hom ðA;BÞ HomomorphismgroupfromanR-moduleAintoan R R-module B HomðG;HÞ Homomorphism group from an Abelian group G into an Abelian group H M(cid:3) N Tensor product of a right R-module M on a left R R-module N K ðRÞ Grothendieck group of the ring R 0 K ðAÞ Grothendieck group of the category offinitely 0 A-projective modules K ðRÞ Whitehead group of the ring R 1 Chapter 1 Introduction Matricesplayanimportantroleinbothpureandappliedmathematics.Matriceswith entriesinringshavebeenintensivelystudiedandused(e.g.,see[22,87]);aswellas matriceswithentriesinsemirings(e.g.,see[39]),Booleanalgebras(e.g.,see[66]), semigroupsandlattices.Thisbookfocusesonamoregeneralobject:formalmatrices or,alternatively,generalizedmatrices.Whatisaformalmatrix?Withoutgoinginto muchdetail,onemightsaythatitisamatrixwithelementsinseveraldifferentrings orbimodules. In a well-known paper [89], Morita introduced objects which are now called Moritacontextsorpre-equivalencesituations.AMoritacontext(R,S,M,N,ϕ,ψ) consistsoftworingsR,SandtwobimodulesM,N whicharelinkedinacertainway bytwobimodulehomomorphismsϕandψ.Initially,Moritacontextswereintroduced to describe equivalences between the categories of modules. They appeared to be a very convenient tool in the study of properties that transfer from one ring R to anotherringS;e.g.,see[6, 63, 64]. Morita contexts have been studied in numerous papers and books. There exist analoguesofMoritacontextsforsemirings,Hopfalgebrasandquasi-Hopfalgebras, coringsandcategories.Withtheuseof(cid:2)theMor(cid:3)itacontext(R,S,M,N,ϕ,ψ),we R M can naturally construct the matrix ring with ordinary matrix operations. N S ThisringiscalledaringofaMoritacontext,oraformalmatrixring(oforder2), oraringofgeneralizedmatrices.Theapproachtakeninthisbookistoconsidera Moritacontextasamatrixring.Itisstraightforwardtodefineaformalmatrixring ofanyordern. Formal matrix rings regularly appear in the study of rings and modules, finite- dimensionalalgebras(e.g.,see[10,11])andendomorphismringsofAbeliangroups (e.g.,see[73]).Theyplayanimportantroleinrealanalysis,e.g.,inconnectionwith operatoralgebras(e.g.,see[17, 31]). Animportantspecialcaseofformalmatrixringsisprovidedbyringsoftriangular matrices or, more generally, rings of block triangular matrices. Rings of (block) ©SpringerInternationalPublishingAG2017 1 P.KrylovandA.Tuganbaev,FormalMatrices, AlgebraandApplications23,DOI10.1007/978-3-319-53907-2_1