Exercises in Basic Ring Theory

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Springer Science & Business Media, Mar 9, 2013 - Mathematics - 200 pages
Each undergraduate course of algebra begins with basic notions and results concerning groups, rings, modules and linear algebra. That is, it begins with simple notions and simple results. Our intention was to provide a collection of exercises which cover only the easy part of ring theory, what we have named the "Basics of Ring Theory". This seems to be the part each student or beginner in ring theory (or even algebra) should know - but surely trying to solve as many of these exercises as possible independently. As difficult (or impossible) as this may seem, we have made every effort to avoid modules, lattices and field extensions in this collection and to remain in the ring area as much as possible. A brief look at the bibliography obviously shows that we don't claim much originality (one could name this the folklore of ring theory) for the statements of the exercises we have chosen (but this was a difficult task: indeed, the 28 titles contain approximatively 15.000 problems and our collection contains only 346). The real value of our book is the part which contains all the solutions of these exercises. We have tried to draw up these solutions as detailed as possible, so that each beginner can progress without skilled help. The book is divided in two parts each consisting of seventeen chapters, the first part containing the exercises and the second part the solutions.
 

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Contents

Zero Divisors
15
Division Rings
31
Socle and Radical
45
Semisimple Rings
49
Prime Ideals Local Rings
53
Polynomial Rings
59
Rings of Quotients
63
Rings of Continuous Functions
67
Ring Homomorphisms
107
Characteristics
111
Divisibility in Integral Domains
115
Division Rings
121
Automorphims
127
The Tensor Product
133
Artinian and Noetherian Rings
139
Socle and Radical
145

Special Problems
73
SOLUTIONS
77
Fundamentals
79
Ideals
91
Zero Divisors
101
Semisimple Rings
153
Prime Ideals Local Rings
159
Polynomial Rings
169
Special problems
187
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