Module theory: endomorphism rings and direct sum decompositions in some classes of modules

The purpose of this expository monograph is three-fold. First, the solution of a problem posed by Wolfgang Krull in 1932 is presented. 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 answer to a question posed by Warfield in 1975, namely, whether theKrull-Schmidt-Theorem holds for serial modules, is described. Facchini published a negative answer in 1996. The solution to the Warfield problem shows an interesting behavior; in fact, it is a phenomenon 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 ringsand modules, exchange property, (sigma)-pure-injective modules). Open problems conclude the work. INDICE: Preface.- List of Symbols.- 1 Basic Concepts.- 2 The Krull-Schmidt-Remak-Azumaya Theorem.- 3 Semiperfect Rings.- 4 Semilocal Rings.- 5 Serial Rings.- 6 Quotient Rings.- 7 Krull Dimension and Serial Rings.- 8 Krull-Schmidt Fails for Finitely Generated Modules and Artinian Modules.- 9 Biuniform Modules.- 10 (Sigma)-pure-injective Modules and Artinian Modules.- 11 Open Problems.- Bibliography.- Index.

• ISBN: 978-3-0348-0302-1
• Editorial: Birkhaüser
• Encuadernacion: Rústica
• Fecha Publicación: 05/02/2012
• Nº Volúmenes: 1
• Idioma: Inglés