LEADER 04979nam 22005415 450 001 9910789344103321 005 20200629114158.0 010 $a1-4612-0923-4 024 7 $a10.1007/978-1-4612-0923-2 035 $a(CKB)3400000000089323 035 $a(SSID)ssj0001295749 035 $a(PQKBManifestationID)11757435 035 $a(PQKBTitleCode)TC0001295749 035 $a(PQKBWorkID)11342822 035 $a(PQKB)11025564 035 $a(DE-He213)978-1-4612-0923-2 035 $a(MiAaPQ)EBC3074626 035 $a(PPN)194113116 035 $a(EXLCZ)993400000000089323 100 $a20121227d1992 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aAlgebra$b[electronic resource] $eAn Approach via Module Theory /$fby William A. Adkins, Steven H. Weintraub 205 $a1st ed. 1992. 210 1$aNew York, NY :$cSpringer New York :$cImprint: Springer,$d1992. 215 $a1 online resource (X, 526 p.) 225 1 $aGraduate Texts in Mathematics,$x0072-5285 ;$v136 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a0-387-97839-9 311 $a1-4612-6948-2 320 $aIncludes bibliographical references and indexes. 327 $a1 Groups -- 1.1 Definitions and Examples -- 1.2 Subgroups and Cosets -- 1.3 Normal Subgroups, Isomorphism Theorems, and Automorphism Groups -- 1.4 Permutation Representations and the Sylow Theorems -- 1.5 The Symmetric Group and Symmetry Groups -- 1.6 Direct and Semidirect Products -- 1.7 Groups of Low Order -- 1.8 Exercises -- 2 Rings -- 2.1 Definitions and Examples -- 2.2 Ideals, Quotient Rings, and Isomorphism Theorems -- 2.3 Quotient Fields and Localization -- 2.4 Polynomial Rings -- 2.5 Principal Ideal Domains and Euclidean Domains -- 2.6 Unique Factorization Domains -- 2.7 Exercises -- 3 Modules and Vector Spaces -- 3.1 Definitions and Examples -- 3.2 Submodules and Quotient Modules -- 3.3 Direct Sums, Exact Sequences, and Horn -- 3.4 Free Modules -- 3.5 Projective Modules -- 3.6 Free Modules over a PID -- 3.7 Finitely Generated Modules over PIDs -- 3.8 Complemented Submodules -- 3.9 Exercises -- 4 Linear Algebra -- 4.1 Matrix Algebra -- 4.2 Determinants and Linear Equations -- 4.3 Matrix Representation of Homomorphisms -- 4.4 Canonical Form Theory -- 4.5 Computational Examples -- 4.6 Inner Product Spaces and Normal Linear Transformations -- 4.7 Exercises -- 5 Matrices over PIDs -- 5.1 Equivalence and Similarity -- 5.2 Hermite Normal Form -- 5.3 Smith Normal Form -- 5.4 Computational Examples -- 5.5 A Rank Criterion for Similarity -- 5.6 Exercises -- 6 Bilinear and Quadratic Forms -- 6.1 Duality -- 6.2 Bilinear and Sesquilinear Forms -- 6.3 Quadratic Forms -- 6.4 Exercises -- 7 Topics in Module Theory -- 7.1 Simple and Semisimple Rings and Modules -- 7.2 Multilinear Algebra -- 7.3 Exercises -- 8 Group Representations -- 8.1 Examples and General Results -- 8.2 Representations of Abelian Groups -- 8.3 Decomposition of the Regular Representation -- 8.4 Characters -- 8.5 Induced Representations -- 8.6 Permutation Representations -- 8.7 Concluding Remarks -- 8.8 Exercises -- Index of Notation -- Index of Terminology. 330 $aThis book is designed as a text for a first-year graduate algebra course. As necessary background we would consider a good undergraduate linear algebra course. An undergraduate abstract algebra course, while helpful, is not necessary (and so an adventurous undergraduate might learn some algebra from this book). Perhaps the principal distinguishing feature of this book is its point of view. Many textbooks tend to be encyclopedic. We have tried to write one that is thematic, with a consistent point of view. The theme, as indicated by our title, is that of modules (though our intention has not been to write a textbook purely on module theory). We begin with some group and ring theory, to set the stage, and then, in the heart of the book, develop module theory. Having developed it, we present some of its applications: canonical forms for linear transformations, bilinear forms, and group representations. Why modules? The answer is that they are a basic unifying concept in mathematics. The reader is probably already familiar with the basic role that vector spaces play in mathematics, and modules are a generaliza­ tion of vector spaces. (To be precise, modules are to rings as vector spaces are to fields. 410 0$aGraduate Texts in Mathematics,$x0072-5285 ;$v136 606 $aAlgebra 606 $aAlgebra$3https://scigraph.springernature.com/ontologies/product-market-codes/M11000 615 0$aAlgebra. 615 14$aAlgebra. 676 $a512 700 $aAdkins$b William A$4aut$4http://id.loc.gov/vocabulary/relators/aut$059612 702 $aWeintraub$b Steven H$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910789344103321 996 $aAlgebra$9382565 997 $aUNINA