Module Theory: Endomorphism Rings & Direct Sum Decompositions in Some Classes of Modules (Progress in Mathematics) - Hardcover

Hardcover

ISBN 10: 0817659080 ISBN 13: 9780817659080

Synopsis

The purpose in writing this expository monograph has been three-fold. First, the author set out to present the solution of a problem posed by Wolfgang Krull in 1932. He asked whether what is now called the "Krull-Schmidt Theorem" holds for artinian modules. A negative answer was published only in 1995 by Facchini, Herbera, Levy and Vámos. Second, the author presents the answer to a question posed by Warfield in 1975, namely, whether the Krull-Schmidt-Theorem holds for serial modules. Facchini published a negative answer in 1996. The solution to the Warfield problem shows an interesting behavior; in fact, it is a phenomena so rare in the history of Krull-Schmidt type theorems that its presentation to a wider mathematical audience provides the third incentive for this monograph. Briefly, the Krull-Schmidt-Theorem holds for some, not all, classes of modules. When it does hold, any two indecomposable decompositions are uniquely determined up to one permutation. For serial modules the theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. Apart from these issues, the book addresses various topics in module theory and ring theory, some now considered classical (such as Goldie dimension, semiperfect rings, Krull dimension, rings of quotients, and their applications) and others more specialized (such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules, exchange property, -pure-injective modules). Open problems conclude the work.

About this title

Review

"Written in an attractive and fresh mathematical style. Each topic is arranged well and lucidly. The author has made important contributions to the study of direct sum decompositions and many of the main results in this book include his own work." --EMS Newsletter

Edition details

• ISBN 10: 0817659080
• ISBN 13: 9780817659080
• Binding: Hardcover
• Language: English
• Number of pages: 285