An Introduction to Rings and Modules: With K-Theory in View
This concise introduction to ring theory, module theory and number theory is ideal for a first year graduate student, as well as being an excellent reference for working mathematicians in other areas. Starting from definitions, the book introduces fundamental constructions of rings and modules, as direct sums or products, and by exact sequences. It then explores the structure of modules over various types of ring: noncommutative polynomial rings, Artinian rings (both semisimple and not), and Dedekind domains. It also shows how Dedekind domains arise in number theory, and explicitly calculates some rings of integers and their class groups. About 200 exercises complement the text and introduce further topics. This book provides the background material for the authors' forthcoming companion volume Categories and Modules. Armed with these two texts, the reader will be ready for more advanced topics in K-theory, homological algebra and algebraic number theory.
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algebraic arbitrary Artinian automorphism bases basis called choose class group clear coefficients commutative composition condition consider construction contains corresponding decomposition Dedekind domain defined definition describe determined direct product direct sum discussion division ring divisor element elementary endomorphism equivalent Euclidean domain example Exercise extension fact field finite fractional ideal free module given gives hence holds homomorphism idempotents identity induction infinite injective integers invertible irreducible isomorphism Lemma matrix maximal minimal module multiplication natural nonzero Note obtain particular polynomial ring prime ideal principal ideal domain projective Proof proper properties Proposition rad(R rank relation respectively result right ideal right Noetherian right R-module scalar short exact sequence Show split standard submodule Suppose surjective Theorem Theorem Let theory twosided ideal unique unit Verify write written zero