Module Theory

This expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the "Krull-Schmidt Theorem" holds for ar tinian modules. The problem remained open for 63 years: its solution, a negative answer to Krull's question, was published only in 1995 (see [Facchini, Herbera, Levy and Vamos]). Secondly, we wanted to present the answer to a question posed by Warfield in 1975 [Warfield 75]. He proved that every finitely pre sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, Warfield asked whether the "Krull-Schmidt Theorem" holds for serial modules. The solution to this problem, a negative answer again, appeared in [Facchini 96]. Thirdly, the so lution to Warfield's problem shows interesting behavior, a rare phenomenon in the history of Krull-Schmidt type theorems. Essentially, the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold, any two indecomposable decompositions are uniquely determined up to a permutation, and when it does not hold for a class of modules, this is proved via an example. For serial modules the Krull-Schmidt Theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phenomenon to a wider math ematical audience.

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.