Algebra : an approach via module theory
著者
書誌事項
Algebra : an approach via module theory
（Graduate texts in mathematics, 136）
Springer-Verlag, c1992
- : us
- : gw
大学図書館所蔵 件 / 全109件
-
該当する所蔵館はありません
- すべての絞り込み条件を解除する
注記
Bibliography: p. [510]
Includes indexes of notation and terminology
内容説明・目次
- 巻冊次
-
: us ISBN 9780387978390
内容説明
This 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.
目次
1 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.
- 巻冊次
-
: gw ISBN 9783540978398
内容説明
This book is designed as a text for a first-year undergraduate-level course in algebra. Instead of an encyclopaedic approach, the authors use a thematic, consistent point of view. The unifying theme is the concept of a module. To promote understanding, the book provides proofs with a maximum of insight and a minimum of computation, as well as chapters stressing computational techniques. Beginning with an introduction to group theory and ring theory, the book then develops the basics of module theory. A review of the basics of linear algebra demonstrates the power of module theory in the derivation of canonical forms and the spectral theorem. The book then uses module theory to investigate bilinear, sesquilinear and quadratic forms. Finally, a discussion of ring theory and multilinear forms leads to a chapter on group representations, which again uses module theory to powerful effect.
目次
1: Groups. 2: Rings. 3: Modules and Vector Spaces. 4: Linear Algebra. 5: Matrices over PIDs. 6: Bilinear and Quadratic Forms. 7: Topics in Module Theory. 8: Group Representations.
「Nielsen BookData」 より