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.

Autorentext

Alberto Facchini studied at the University of Padua, getting the Laurea in Matematica magna cum laude in 1977. He has worked at the University of Udine from 1981 to 1999, becoming full professor of Algebra, and then at the University of Padua from 1999 to 2022. At present he is Emeritus Professor at the University of Padua. He also got fellowships to study and do research at the University of Sheffield, Universitat Autònoma de Barcelona, and Centre de Recerca Matemàtica. He has written about 180 scientific publications and has given lectures and talks at conferences in about forty different countries. In 1981 he received the Bonavera Prize in Mathematics. He has been in the editorial boards of about fifteen mathematical journal, among which Bollettino dell'Unione Matematica Italiana, Communications in Algebra, Journal of Algebra and Its Applications, Rendiconti del Circolo Matematico di Palermo, and Rendiconti del Seminario Matematico dell'Università di Padova.

Inhalt
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.