The Concise Handbook of Algebra by

Alexander V. Mikhalev Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

and

Gunter F. Pilz Institute of Algebra,, Johannes Kepler University-Linz, Linz, Austria

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Contents

Preface A SEMIGROUPS A.I Ideals and Green's Relations by Lev N. Shevrin in Ekaterinburg, Russia A.2 Bands of Semigroups by Lev N. Shevrin in Ekaterinburg, Russia A.3 Free Semigroups by Lev N. Shevrin in Ekaterinburg, Russia A.4 Presentations and Word Problems by Peter M. Higgins in Colchester and Nik Ruskuc in St. Andrews, UK A.5 Simple Semigroups by Lev N. Shevrin in Ekaterinburg, Russia A.6 Epigroups by Lev N. Shevrin in Ekaterinburg, Russia A.7 Periodic Semigroups by Lev N. Shevrin in Ekaterinburg, Russia A.8 Finite Semigroups and Pseudovarieties by Jorge Almeida in Porto, Portugal A.9 Regular Semigroups by Peter G. Trotter in Hobart, Australia A. 10 Completely Regular Semigroups by John M. Howie in St. Andrews, U.K A. 11 Inverse Semigroups by Peter G. Trotter in Hobart, Australia A. 12 Separated Transformation Semigroups by Kenneth D. Magill, Jr. in Buffalo, NY, USA A. 13 Matrix Semigroups by Jan Okniriski in Warsaw, Poland

xv 1 1 5 8

13 17 23 28 31 35 39 42 46 49

vi

CONTENTS A. 14 Subsemigroup Lattices by A. J. Ovsyannikov and Lev N. Shevrin in Ekaterinburg, Russia A.15 Varieties of Semigroups by Lev N. Shevrin and Mikhail V. Volkov in Ekaterinburg, Russia A. 16 Compact Semigroups by Karl H. Hofmann in Darmstadt, Germany A.17 Applications of Semigroups by Gu'nter F. Pilz in Linz, Austria, Lev N. Shevrin in Ekaterinburg, Russia, and Peter G. Trotter in Hobart, Australia

53

58 62

66

B GROUPS 71 B.I Abelian Groups by Alexander V. Mikhalev in Moscow, Russia 71 B.2 p-Groups: Basics and the Coclass Project by Charles R. Leedham-Green in London, UK 75 B.3 p-Groups: Other Approaches by Charles R. Leedham-Green in London, UK 79 B.4 Soluble Groups by Hans Lausch in Melbourne, Australia 82 B.5 Permutation Groups by Peter J. Cameron in London, UK 86 B.6 Lie Groups by Joachim Hilgert in Clausthal, Germany 89 B.7 Lie Groups and Differential Equations by Peter J. Olver in Minneapolis, MN, USA 92 B.8 Simple Groups by Ronald Solomon in Columbus, OH, USA 97 B.9 Free and Relatively Free Groups by Gilbert Baumslag in New York, NY, USA 102 B.10 Free Products, Amalgamated Products, and HNN Extensions by Gilbert Baumslag in New York, NY, USA 104 B.ll Word Problems in Groups by Derek Holt in Warwick, UK 107 B.12 Combinatorial Group Theory by Gilbert Baumslag in New York, NY, USA . . 109

CONTENTS

vii

B.13 The Burnside Problems by Rostislav I. Grigorchuk and Igor G. Lysenok in Moscow, Russia Ill B.14 Automatic and Hyperbolic Groups by Derek Holt in Warwick, UK 115 B.I5 Actions of Finite Groups by Peter Paule in Linz, Austria 119 B.16 Wreath Products by John D. P.Meldrum in Edinburgh, UK 123 B.17 Automorphism Groups of Algebraic Systems by Boris I. Plotkin in Jerusalem, Israel 127 B.18 Frobenius Groups by Peter Fleischmann in Kent, UK 131 B.19 Covered and Fibered Groups by C a r l t o n J. M a x s o n i n College

S t a t i o n , TX, USA . . . .

B.20 Subgroup Lattices of Finite Abelian Groups by Frank Vogt in Darmstadt, Germany B.21 Varieties of Groups by Awad A. Iskander in Lafayette, LA, USA B.22 Groups and Absolute Geometry by Helmut Karzel in Munich, Germany B.23 Groups for Particle Physics by Gudrun Kalmbach H. E. in Ulm, Germany C RINGS, MODULES, ALGEBRAS C.I Commutative Algebra I: Ideal Decompositions by Robert Gilmer in Tallahassee, FL, USA C.2 Commutative Algebra II: The Krull Dimension by Robert Gilmer in Tallahassee, FL, USA C.3 Factorization and Decomposition of Polynomials by Joachim von zur Gathen in Paderborn, Germany . . C.4 Element Factorization in Integral Domains by David F. Anderson in Knoxville, TN, USA C.5 Dedekind and Priifer Domains by Marco Fontana in Rome, Italy and Ira J. Papick in Columbia, MO, USA • C.6 Local Rings by T. Y. Lam in Berkeley, CA, USA C.7 Semilocal Rings by T. Y. Lam in Berkeley, CA, USA

134

137 140 144 148 153 153 156 . 159 161

165 169 173

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CONTENTS C.8 Cohen-Macaulay Rings by Ju'rgen Herzog in Essen, Germany C.9 Rings of Formal Power Series by Askar A. Tuganbaev in Moscow, Russia CIO Automorphisms of Polynomial Algebras and the Jacobian Conjecture by Jie-Tai Yu in Hong Kong, China C.ll Rings of Integer-Valued Polynomials by Paul-Jean Cohen in Marseille and Jean-Luc Chabert in Amiens, France C.12 Commutativity Conditions by Howard E. Bell in St. Catharines, Canada C.13 Finite Dimensional Division Algebras by David Saltman in Austin, TX, USA C.14 Infinite Dimensional Division Rings by Paul M. Cohn in London, UK C.I5 Rings of Quotients by Wallace S . M a r t i n d a l e , 3 r d , i n G l e n s i d e , P A , USA . . . C.16 Endomorphism and Matrix Rings by Alexander V. Mikhalev and Askar A. Tuganbaev in Moscow, Russia C.I7 Primitive Rings and Semisimplicity by Kostia Beidar in Tainan, Taiwan C.I8 The Jacobson Radical by Richard Wiegandt in Budapest, Hungary C.19 General Radical Theory by Richard Wiegandt in Budapest, Hungary C.20 Free and Projective Modules by Alexander V. Mikhalev and Askar A. Tuganbaev in Moscow, Russia C.21 Injective Modules by Askar A. Tuganbaev in Moscow, Russia C.22 Flat Modules by Askar A. Tuganbaev in Moscow, Russia C.23 Free Ideal Rings and Free Algebras by Paul M. Cohn in London, UK C.24 Simple Rings by Kostia Beidar in Tainan, Taiwan C.25 Principal Ideal Domains by Paul M. Cohn in London, UK

177 182

185

188 192 193 198 202

205 209 213 215

219 223 227 231 235 239

CONTENTS

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C.26 Noetherian Rings by Lance W. Small in San Diego, CA, USA 242 C.27 Artinian Rings by Victor T. Markov in Moscow, Russia 244 C.28 Maximal, Prime, and Semiprime by Gary F. Birkenmeier in Lafayette, LA, USA 249 C.29 Regular Rings by Alexander V. Mikhalev and Askar A. Tuganbaev in Moscow, Russia 253 C.30 Quasi-Frobenius and Self-injective Rings by Alexander V. Mikhalev and Askar A. Tuganbaev in Moscow, Russia 258 C.31 Grobner Bases: The Commutative Case by Bruno Buchberger in Linz, Austria 262 C.32 Grobner Bases: Applications by Bruno Buchberger in Linz, Austria 265 C.33 Grobner Bases: The Non-commutative Case by Leonid Bokut' in Novosibirsk, Russia 268 C.34 Orders and Maximal Orders by T. Y. Lam in Berkeley, CA, USA 273 C.35 Representations of Orders by Vladimir Kirichenko in Kiev, Ukraine 278 C.36 Distributive Modules and Rings by Askar A. Tuganbaev in Moscow, Russia 280 C.37 Pi-Algebras by Alexei J. Belov in Perth, Australia, and Louis H. Rowen in Ramat Gan, Israel 284 C.38 Clifford Algebras by Jaques Helmstetter in Grenoble, France 289 C.39 Rings with Involution by Henry E. Heatherly in Lafayette, LA, USA 293 C.40 Rings of Operators by Henry E. Heatherly in Lafayette, LA, USA 296 C.41 The Gel'fand-Kirillov Dimension by Vesselin Drensky in Sofia, Bulgaria 300 C.42 Coalgebras by Vyacheslav A. Artamonov in Moscow, Russia 305 C.43 Hopf Algebras by Vyacheslav A. Artamonov in Moscow, Russia 308

x

CONTENTS C.44 Ordered Rings by Laszlo Fuchs in New Orleans, LA, USA 311 C.45 Topological Rings and Modules by S.T.Glavatsky and A. V.Mikhalev in Moscow, Russia . 314 C.46 Semirings and Semifields by Udo Hebisch in Freiberg and Hanns J. Weinert in Clausthal, Germany 318 C.47 Near-Rings and Near-Fields by Gu'nter F. Pilz in Linz, Austria 322 C.48 Conformal Algebras by Efim Zelmanov in New Haven, CT, USA 326 C.49 Lie Algebras by Mikhael Zaicev in Moscow, Russia 330 C.50 Simple Lie Algebras by Andrej A. Zolotykh in Moscow, Russia 333 C.51 Varieties of Lie Algebras by Mikhael Zaicev in Moscow, Russia 337 C.52 Lie Superalgebras by Alexander A. Mikhalev in Moscow, Russia 339 C.53 Other Nonassociative Algebras by Luiz A. Peresi in Sao Paulo, Brazil 343 C.54 Computational Ring Theory by Franz Winkler in Linz, Austria 347 C.55 Applications of Rings by Gu'nter F. Pilz in Linz, Austria 350

D FIELDS

355

D.I Field Extensions by Shreeram S. Abhyankar in Lafayette, IN, USA 355 D.2 Finite Fields by Harald Niederreiter in Singapore 359 D.3 Galois Theory by Joseph J. Rotman in Urbana, IL, USA 362 D.4 Differential Galois Theory by Andy R. Magid in Norman, OK, USA 366 D.5 Differential Dimension Polynomials by A. V. Mikhalev and E. V. Pankratiev in Moscow, Russia369 D.6 Ordered Fields by T. Y. Lam in Berkeley, CA, USA 373

CONTENTS

xi

D.7 Applications of Fields by Franz Binder in Linz, Austria and Rudi Lidl in Hobart, Australia 377 E REPRESENTATION THEORY 383 E.I Group Rings by Donald S. -Passman in Madison, WI, USA 383 E.2 Character Theory by I. Martin Isaacs in Madison, WI, USA 386 E.3 Ordinary Representations I by Albert Fdssler in Biel, Switzerland 390 E.4 Ordinary Representations II by Albert Fdssler in Biel, Switzerland 392 E.5 Modular Representations: General theory by Peter Fleischmann in Kent, UK 398 E.6 Modular Representations: Brauer's Theorems by Peter Fleischmann in Kent, UK 400 E.7 Characters of the Symmetric and Alternating Groups by Albert Fdssler in Biel, Switzerland 403 E.8 Representation Theory of Lie Groups by Joachim Hilgert in Clausthal, Germany 405 E.9 Groups in 2D/3D vision problems by Kenichi Kanatani in Okayama, Japan and Gu'nter F. Pilz in Linz, Austria 409 E.10 Representation Theory and Statistics by Albert Fdssler in Biel, Switzerland 412 F LATTICES , 417 F.I Congruences and Constructions by George Grdtzer in Winnipeg, Canada and E. Tamds Schmidt in Budapest, Hungary 417 F.2 Modular Lattices by Ralph Freese in Honolulu, HI, USA 420 F.3 Distributive Lattices; Heyting and Post Algebras by Viacheslav N. Salii in Saratov (Russia) 423 F.4 Complemented and Orthocomplemented Lattices by Gudrun Kalmbach H. E. in Ulm, Germany 426 F.5 Orthomodular Lattices by Gudrun Kalmbach H. E. in Ulm, Germany 429

xii

CONTENTS F.6 Boolean Algebras by Carlton J. Maxson in College Station, TX, USA and Gu'nter F. Pilz in Linz, Austria F.7 Complete Lattices by Viacheslav N. Salii in Saratov, Russia F.8 Free Lattices by Ralph Freese in Honolulu, HI, USA F.9 Varieties of Lattices by George Grdtzer in Winnipeg, Canada F.10 Applications of Lattices: Formal Concept Analysis by Rudolf Wille in Darmstadt, Germany

432 436 439 442 446

G UNIVERSAL ALGEBRA 451 G.I Constructions in Universal Algebras by Vyacheslav A. Artamonov in Moscow, Russia, and Gu'nter F. Pilz in Linz, Austria 451 G.2 Automorphisms and Endomorphisms in Universal Algebra by Boris I. Plotkin in Jerusalem, Israel 455 G.3 Polynomials and Polynomial Completeness by Kalle Kaarli in Tartu, Estonia 457 G.4 Free Algebras by Lev N. Shevrin and Evgeny V. Sukhanov in Ekaterinburg, Russia 461 G.5 Varieties and Quasi-varieties by Lev N. Shevrin and Mikhail V. Volkov in Ekaterinburg, Russia 465 G.6 Congruence Modular Varieties by A. G. Pinus in Novosibirsk, Russia and Yefim Katsov in Hanover, IN, USA ' . . . 469 G.7 Word Problems and Rewriting Systems by Leonid Bokut' in Novosibirsk, Russia 474 G.8 Relational Algebras by Hajnal Andreka, Judit X. Madardsz, and Istvdn Nemeti in Budapest, Hungary 478 G.9 Partial Algebras by Peter Burmeister in Darmstadt, Germany 482 G.10 Abstract Data Types by H a n s - D i e t e r E h r i c h i n B r a u n s c h w e i g , G e r m a n y

....

486

CONTENTS

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H HOMOLOGICAL ALGEBRA H.I Foundations of Homological Algebra by Peter Hilton in Binghampton, NY, USA H.2 Universal Constructions by Yefim Katsov in Hanover, IN, USA H.3 Projective and Injective Resolutions by Yefim Katsov in Hanover, IN, USA H.4 Homological Dimension by Peter Hilton in Binghampton, NY, USA H.5 Homological Characterizations of Rings: The Commutative Case by Sarah Glaz in Storrs, CT, USA H.6 Homological Characterizations of Rings: The General Case by Alexander V. Mikhalev and Askar A. Tuganbaev in Moscow, Russia H.7 Algebraic if-Theory I: Ko and Kx by Jonathan M. Rosenberg in College Park, MD, USA H.8 Algebraic if-theory II: K2, Milnor if-theory, and Symbols by Jonathan M. Rosenberg in College Park, MD, USA H.9 Algebraic if-Theory III: Higher Algebraic X-Theory by Jonathan M. Rosenberg in College Park, MD, USA

491 491 493 497 502 505

508 . . 513 . . 517 . . 520

I MISCELLANEOUS 1.1 Model Theory for Algebraists by H. Jerome Keisler in Madison, WI, USA 1.2 Linear Codes over Fields by Gu'nter F. Pilz in Linz, Austria 1.3 Linear Codes over Rings by V. T. Markov, A. V. Mikhalev, and A. A. Nechaev in Moscow, Russia 1.4 History of Algebra before 1500 by Hans Kaiser in Vienna, Austria 1.5 History of Algebra after 1500 by John Stillwell in Melbourne, Australia

523

Bibliography

543

About the Authors

585

Index

595

523 526 530 535 539