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 the Krull-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 phenomena so rare…mehr
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 the Krull-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 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, (sigma)-pure-injective modules). Open problems conclude the work.
Besides its research value, this book is a considerable addition to the list of fundamental books in rings and modules. It is written carefully with necessary backgrounds developed in a logical way. There are many fundamental points worked out in the book. All these make the book an excellent contribution to the development of module and ring theory, and a source of reference. Algebraists will certainly enjoy themselves in reading this book.
This excellent book on module and ring theory (...) contains three kind of topics: classical topics, (...) specialized topics, (...) the solutions to two famous problems, together with the presentation of a rare phenomenon.
Alberto Facchini is a Professor of Mathematics at the University of Padua.
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.
1 Basic Concepts.- 1.1 Semisimple rings and modules.- 1.2 Local and semilocal rings.- 1.3 Serial rings and modules.- 1.4 Pure exact sequences.- 1.5 Finitely definable subgroups and pure-injective modules.- 1.6 The category (RFP, Ab).- 1.7 ?-pure-injective modules.- 1.8 Notes on Chapter 1.- 2 The Krull-Schmidt-Remak-Azumaya Theorem.- 2.1 The exchange property.- 2.2 Indecomposable modules with the exchange property.- 2.3 Isomorphic refinements of finite direct sum decompositions.- 2.4 The Krull-Schmidt-Remak-Azumaya Theorem.- 2.5 Applications.- 2.6 Goldie dimension of a modular lattice.- 2.7 Goldie dimension of a module.- 2.8 Dual Goldie dimension of a module.- 2.9 ?-small modules and ?-closed classes.- 2.10 Direct sums of ?-small modules.- 2.11 The Loewy series.- 2.12 Artinian right modules over commutative or right noetherian rings.- 2.13 Notes on Chapter 2.- 3 Semiperfect Rings.- 3.1 Projective covers and lifting idempotents.- 3.2 Semiperfect rings.- 3.3 Modules over semiperfect rings.- 3.4 Finitely presented and Fitting modules.- 3.5 Finitely presented modules over serial rings.- 3.6 Notes on Chapter 3.- 4 Semilocal Rings.- 4.1 The Camps-Dicks Theorem.- 4.2 Modules with semilocal endomorphism ring.- 4.3 Examples.- 4.4 Notes on Chapter 4.- 5 Serial Rings.- 5.1 Chain rings and right chain rings.- 5.2 Modules over artinian serial rings.- 5.3 Nonsingular and semihereditary serial rings.- 5.4 Noetherian serial rings.- 5.5 Notes on Chapter 5.- 6 Quotient Rings.- 6.1 Quotient rings of arbitrary rings.- 6.2 Nil subrings of right Goldie rings.- 6.3 Reduced rank.- 6.4 Localization in chain rings.- 6.5 Localizable systems in a serial ring.- 6.6 An example.- 6.7 Prime ideals in serial rings.- 6.8 Goldie semiprime ideals.- 6.9 Diagonalization of matrices.- 6.10 Ore sets in serial rings.- 6.11 Goldie semiprime ideals and maximal Ore sets.- 6.12 Classical quotient ring of a serial ring.- 6.13 Notes on Chapter 6.- 7 Krull Dimension and Serial Rings.- 7.1 Deviation of a poset.- 7.2 Krull dimension of arbitrary modules and rings.- 7.3 Nil subrings of rings with right Krull dimension.- 7.4 Transfinite powers of the Jacobson radical.- 7.5 Structure of serial rings of finite Krull dimension.- 7.6 Notes on Chapter 7.- 8 Krull-Schmidt Fails for Finitely Generated Modules and Artinian Modules.- 8.1 Krull-Schmidt fails for finitely generated modules.- 8.2 Krull-Schmidt fails for artinian modules.- 8.3 Notes on Chapter 8.- 9 Biuniform Modules.- 9.1 First properties of biuniform modules.- 9.2 Some technical lemmas.- 9.3 A sufficient condition.- 9.4 Weak Krull-Schmidt Theorem for biuniform modules.- 9.5 Krull-Schmidt holds for finitely presented modules over chain rings.- 9.6 Krull-Schmidt fails for finitely presented modules over serial rings.- 9.7 Further examples of biuniform modules of type 1.- 9.8 Quasi-small uniserial modules.- 9.9 A necessary condition for families of uniserial modules.- 9.10 Notes on Chapter 9.- 10 ?-pure-injective Modules and Artinian Modules.- 10.1 Rings with a faithful ?-pure-injective module.- 10.2 Rings isomorphic to endomorphism rings of artinian modules.- 10.3 Distributive modules.- 10.4 ?-pure-injective modules over chain rings.- 10.5 Homogeneous ?-pure-injective modules.- 10.6 Krull dimension and ?-pure-injective modules.- 10.7 Serial rings that are endomorphism rings of artinian modules.- 10.8 Localizable systems and ?-pure-injective modules over serial rings.- 10.9 Notes on Chapter 10.- 11 Open Problems.
"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
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