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**Exercises in classical ring theory.**
*(English)*
Zbl 0823.16001

Problem Books in Mathematics. Berlin: Springer-Verlag. xvi, 287 p. (1994).

The book contains solutions to 400 ring-theoretic exercises and will be very useful for everybody studying ring theory.

From the preface: “There are many good reasons for a problem book to be written in ring theory, or for that matter in any subject of mathematics. First, the solutions to different exercises serve to illustrate the problem-solving process and show how general theorems in ring theory are applied in special situations. Second, the compilation of solutions to interesting and unusual exercises extends and completes the standard treatment of the subject in textbooks. Last, but not least, a problem book provides a natural place in which to record leisurely some of the folklore of the subject: the “tricks of the trade” in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference. …For the reader’s convenience, each section in this book begins with a short introduction giving the general background and the theoretical basis for the problems that follow. All problems are solved in full, in most cases with a lot more details than can be found in the original sources (if they exist). A majority of the solutions are accompanied by a “Comment” section giving relevant bibliographical, historical or anecdotical information.”

Contents: Preface; Notes to the reader; Chapter 1 Wedderburn-Artin theory (59 exercises); Chapter 2 Jacobson radical theory (55 exercises); Chapter 3 Introduction to representation theory (43 exercises); Chapter 4 Prime and primitive rings (74 exercises); Chapter 5 Introduction to division rings (55 exercises); Chapter 6 Ordered structures in rings (22 exercises); Chapter 7 Local rings, semilocal rings, and idempotents (72 exercises); Chapter 8 Perfect and semiperfect rings (20 exercises); Name index; Subject index.

From the preface: “There are many good reasons for a problem book to be written in ring theory, or for that matter in any subject of mathematics. First, the solutions to different exercises serve to illustrate the problem-solving process and show how general theorems in ring theory are applied in special situations. Second, the compilation of solutions to interesting and unusual exercises extends and completes the standard treatment of the subject in textbooks. Last, but not least, a problem book provides a natural place in which to record leisurely some of the folklore of the subject: the “tricks of the trade” in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference. …For the reader’s convenience, each section in this book begins with a short introduction giving the general background and the theoretical basis for the problems that follow. All problems are solved in full, in most cases with a lot more details than can be found in the original sources (if they exist). A majority of the solutions are accompanied by a “Comment” section giving relevant bibliographical, historical or anecdotical information.”

Contents: Preface; Notes to the reader; Chapter 1 Wedderburn-Artin theory (59 exercises); Chapter 2 Jacobson radical theory (55 exercises); Chapter 3 Introduction to representation theory (43 exercises); Chapter 4 Prime and primitive rings (74 exercises); Chapter 5 Introduction to division rings (55 exercises); Chapter 6 Ordered structures in rings (22 exercises); Chapter 7 Local rings, semilocal rings, and idempotents (72 exercises); Chapter 8 Perfect and semiperfect rings (20 exercises); Name index; Subject index.

Reviewer: J.Ponizovskij (St.Peterburg)

### MSC:

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

00A07 | Problem books |

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\textit{T. Y. Lam}, Exercises in classical ring theory. Berlin: Springer-Verlag (1994; Zbl 0823.16001)