Exercises in Basic Ring Theory / Edition 1

by Grigore Calugareanu, P. Hamburg

Overview

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.

Product Details

ISBN-13:	9780792349181
Publisher:	Springer Netherlands
Publication date:	02/28/1998
Series:	Texts in the Mathematical Sciences , #20
Edition description:	1998
Pages:	200
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

I Exercises.- 1 Fundamentals.- 2 Ideals.- 3 Zero Divisors.- 4 Ring Homomorphisms.- 5 Characteristics.- 6 Divisibility in Integral Domains.- 7 Division Rings.- 8 Automorphisms.- 9 The Tensor Product.- 10 Artinian and Noetherian Rings.- 11 Socle and Radical.- 12 Semisimple Rings.- 13 Prime Ideals, Local Rings.- 14 Polynomial Rings.- 15 Rings of Quotients.- 16 Rings of Continuous Functions.- 17 Special Problems.- II Solutions.- 1 Fundamentals.- 2 Ideals.- 3 Zero Divisors.- 4 Ring Homomorphisms.- 5 Characteristics.- 6 Divisibility in Integral Domains.- 7 Division Rings.- 8 Automorphims.- 9 The Tensor Product.- 10 Artinian and Noetherian Rings.- 11 Socle and Radical.- 12 Semisimple Rings.- 13 Prime Ideals, Local Rings.- 14 Polynomial Rings.- 15 Rings of Quotients.- 16 Rings of Continuous Functions.- 17 Special problems.

Editorial Reviews

Offers undergraduate students 346 exercises using and demonstrating the elementary realms of ring theory, along with introductions to each chapter explaining the basic definitions and results, and the solutions to each exercise. Assumes a knowledge of basic facts concerning groups and modules, such as cyclic groups and the Lagrange theorem. Includes such topics as fundamentals, ring homomorphisms, division rings, the tensor product, socle and radical, polynomial rings, rings of continuous functions, and special problems. Annotation c. by Book News, Inc., Portland, Or.

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