Verfügbarkeit: Introduction to noncommutative algebra

Introduction to noncommutative algebra:

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1. Verfasser:	Brešar, Matej (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Cham [u.a.] Springer 2014
Schriftenreihe:	Universitext
Online-Zugang:	Inhaltsverzeichnis Klappentext
Beschreibung:	Includes bibliographical references and index
Beschreibung:	XXXVII, 199 S. 24 cm
ISBN:	3319086928 9783319086927

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MARC


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Datensatz im Suchindex

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adam_text	Contents Finite Dimensional Division Algebras ...................... 1 1.1 After the Complex Numbers: What Comes Next? .......... 1 1.2 Beyond Frobenius Theorem ......................... 4 1.3 Simple Rings .................................... 6 1.4 Central Algebras ................................. 9 1.5 Multiplication Algebra ............................. 11 1.6 Automorphisms of Central Simple Algebras .............. 13 1.7 Maximal Subfields ................................ 15 1.8 Wedderburn s Theorem on Finite Division Rings ........... 16 1.9 Further Examples of Division Algebras ................. 18 Exercises ........................................... 21 Structure of Finite Dimensional Algebras ................... 25 2.1 Nilpotent Ideals .................................. 25 2.2 Prime and Semiprime Rings ......................... 28 2.3 Unitization ..................................... 31 2.4 The Regular Representation .......................... 32 2.5 Group Algebras .................................. 35 2.6 Matrix Units .................................... 37 2.7 Idempotents ..................................... 39 2.8 Minimal Left Ideals ............................... 41 2.9 Wedderburn s Structure Theorems ..................... 42 2.10 Algebras Over Special Fields ........................ 45 2.11 Scalar Extension (A Naive Approach) .................. 47 Exercises ........................................... 49 Modules and Vector Spaces ............................. 53 3.1 Concept of a Module .............................. 53 3.2 Basic Module-Theoretic Notions ...................... 56 3.3 Vector Spaces Over Division Rings .................... 59 3.4 Endomorphisms and Matrices ........................ 61 Contents 3.5 Simple Modules.................................. W 3.6 Maximal Left Ideals............................... 65 3.7 Schur s Lemma.................................. 67 3.8 Semisimple Modules.............................. 68 3.9 Wedderburn s Structure Theory Revisited ................ 69 ЗЛО Chain Conditions ................................. 72 ■ * · f s Exercises ........................................... 79 Tensor Products ...................................... 79 4.1 Concept of a Tensor Product ......................... 4.2 Basic Properties of Tensor Products .................... 4.3 Linear (In)dependence in Tensor Products ............... 84 4.4 Tensor Product of Algebras .......................... 4.5 Multiplication Algebra and Tensor Products .............. 9l 4.6 Centralizers in Tensor Products ....................... l¿ 4.7 Scalar Extension (The Right Approach) ............... 94 4.8 Simplicity of Tensor Products ........................ 4.9 The Skolem-Noether Theorem ........................ lb 99 4.10 The Double Centralizer Theorem ...................... 4.11 The Brauer Group ................................ 101 Exercises ........................................... Structure of Rings .................................... l07 5.1 Primitive Rings .................................. l07 5.2 The Jacobson Density Theorem ....................... 5.3 Alternative Versions and Applications .................. 5.4 Primitive Rings Having Minimal Left Ideals .............. ** 5.5 Primitive Ideals .................................. 119 5.6 Introducing the Jacobson Radical ...................... 5.7 Quasi-invertibility ................................ 5.8 Computing the Jacobson Radical ...................... 12ţj 5.9 Semiprimitive Rings ............................... ^ 5.10 Structure Theory in Action .......................... 13° Exercises ........................................... 13 Noncommutative Polynomials ............................ 6.1 Free Algebras ................................... ^ 6.2 Algebras Defined by Generators and Relations ............ 14i^ 6.3 Alternating Polynomials ............................ 14^ 6.4 Polynomial Identities: Definition and Examples ............ l45 6.5 Linearization .................................... ^47 6.6 Stable Identities .................................. ł49 6.7 T-ideals ........ . ............................ 152 Contents xiii 6.8 The Characteristic Polynomial ........................ 153 6.9 The Amitsur-Levitzki Theorem ....................... 157 Exercises ........................................... 159 7 Rings of Quotients and Structure of Pi-Rings ................ 163 7.1 Rings of Central Quotients .......................... 163 7.2 Classical Rings of Quotients ......................... 166 7.3 Ore Domains .................................... 169 7.4 Martindale Rings of Quotients ........................ 171 7.5 The Extended Centroid ............................. 174 7.6 Linear (Independence in Prime Rings .................. 177 7.7 Prime GPI-Rings ................................. 179 7.8 Primitive Pi-Rings ................................ 183 7.9 Prime Pi-Rings .................................. 185 7.10 Central Polynomials ............................... 188 Exercises ........................................... 189 References ............................................ 193 Index ................................................ 195 Universitext Matej Brešar Introduction to Noncommutative Algebra Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson s structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.
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spelling	Brešar, Matej Verfasser aut Introduction to noncommutative algebra Matej Brešar Cham [u.a.] Springer 2014 XXXVII, 199 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Universitext Includes bibliographical references and index Erscheint auch als Online-Ausgabe 978-3-319-08693-4 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615535&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615535&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext
spellingShingle	Brešar, Matej Introduction to noncommutative algebra
title	Introduction to noncommutative algebra
title_auth	Introduction to noncommutative algebra
title_exact_search	Introduction to noncommutative algebra
title_full	Introduction to noncommutative algebra Matej Brešar
title_fullStr	Introduction to noncommutative algebra Matej Brešar
title_full_unstemmed	Introduction to noncommutative algebra Matej Brešar
title_short	Introduction to noncommutative algebra
title_sort	introduction to noncommutative algebra
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615535&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615535&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
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