Exercises in Classical Ring Theory
Springer Science & Business Media, 29 juin 2013 - 288 pages
Based in large part on the comprehensive "First Course in Ring Theory" by the same author, this book provides a comprehensive set of problems and solutions in ring theory that will serve not only as a teaching aid to instructors using that book, but also for students, who will see how ring theory theorems are applied to solving ring-theoretic problems and how good proofs are written.
The author demonstrates that problem-solving is a lively process: in "Comments" following many solutions he discusses what happens if a hypothesis is removed, whether the exercise can be further generalized, what would be a concrete example for the exercise, and so forth. The book is thus much more than a solution manual.
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Introduction to Division Rings
15 Tensor products and maximal subfields
Ordered Structures in Rings
Local Rings Semilocal Rings
21 The theory of idempotents
22 Central idempotents and block decompositions
24 Homological characterizations of perfect
12 Subdirect products and commutativity theorems
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acts algebra artinian assume central character claim clearly commutative conjugate consider constructed contains contradiction Conversely course defined desired direct division ring domain easy element equation equivalent example Exercise exists extension fact field finite follows formally given gives group G hence holds homomorphism idempotent identity implies infinite instance integer inverse irreducible isomorphism left primitive left R-module linear matrix maximal left ideal means minimal module multiplication Neumann regular Nilº nilpotent noetherian nonzero Note particular polynomial problem projective proof properties prove rad kG radical reduced relations representation resp result right ideal root satisfies semilocal semiprime semisimple semisimple ring Show simple Solution strongly subgroup submodule subring suffices Suppose Theorem theory unique unit write zero