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Full Description
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.
Contents
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.
