Module Theory

"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

• 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 inserial 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.