Algebra [electronic resource] : A Teaching and Source Book / by Ernest Shult, David Surowski.
Contributor(s): Surowski, David [author.] | SpringerLink (Online service)Material type: TextPublisher: Cham : Springer International Publishing : Imprint: Springer, 2015Description: XXII, 539 p. 6 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783319197340Subject(s): Mathematics | Algebra | Associative rings | Rings (Algebra) | Field theory (Physics) | Group theory | Mathematics | Associative Rings and Algebras | Group Theory and Generalizations | Field Theory and Polynomials | AlgebraAdditional physical formats: Printed edition:: No titleDDC classification: 512.46 LOC classification: QA251.5Online resources: Click here to access online
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Basics -- Basic Combinatorial Principles of Algebra -- Review of Elementary Group Properties -- Permutation Groups and Group Actions -- Normal Structure of Groups -- Generation in Groups -- Elementary Properties of Rings -- Elementary properties of Modules -- The Arithmetic of Integral Domains -- Principal Ideal Domains and Their Modules -- Theory of Fields -- Semiprime Rings -- Tensor Products.
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.