Algebra: A Teaching and Source Book

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This book presents a graduate-level course on modern algebra. It can be used as a teaching book – owing to the copious exercises – and as a source book for those who wish to use the major theorems of algebra.

The course begins with the basic combinatorial principles of algebra: posets, chain conditions, Galois connections, and dependence theories. Here, the general Jordan–Holder Theorem becomes a theorem on interval measures of certain lower semilattices. This is followed by basic courses on groups, rings and modules; the arithmetic of integral domains; fields; the categorical point of view; and tensor products.

Beginning with introductory concepts and examples, each chapter proceeds gradually towards its more complex theorems. Proofs progress step-by-step from first principles. Many interesting results reside in the exercises, for example, the proof that ideals in a Dedekind domain are generated by at most two elements. The emphasis throughout is on real understanding as opposed to memorizing a catechism and so some chapters offer curiosity-driven appendices for the self-motivated student.

Author(s): Ernest Shult, David Surowski (auth.)
Edition: 1
Publisher: Springer International Publishing
Year: 2015

Language: English
Pages: XXII, 539
Tags: Associative Rings and Algebras; Group Theory and Generalizations; Field Theory and Polynomials; Algebra

Front Matter....Pages i-xxii
Basics....Pages 1-19
Basic Combinatorial Principles of Algebra....Pages 21-71
Review of Elementary Group Properties....Pages 73-103
Permutation Groups and Group Actions....Pages 105-136
Normal Structure of Groups....Pages 137-161
Generation in Groups....Pages 163-184
Elementary Properties of Rings ....Pages 185-230
Elementary Properties of Modules....Pages 231-277
The Arithmetic of Integral Domains....Pages 279-332
Principal Ideal Domains and Their Modules....Pages 333-354
Theory of Fields....Pages 355-441
Semiprime Rings....Pages 443-469
Tensor Products....Pages 471-527
Back Matter....Pages 529-539