Author | Facchini, Alberto. author |
---|---|

Title | Module Theory [electronic resource] : Endomorphism rings and direct sum decompositions in some classes of modules / by Alberto Facchini |

Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1998 |

Connect to | http://dx.doi.org/10.1007/978-3-0348-8774-8 |

Descript | XIII, 288 p. online resource |

SUMMARY

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 arยญ 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 preยญ 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 soยญ 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 mathยญ ematical audience

CONTENT

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

Mathematics
Algebra
Mathematics
Algebra