Lifting Modules: Supplements and Projectivity in Module Theory

Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. There is a certain asymmetry in this duality. While the theory of extending modules is well documented in monographs and text books, the purpose of our monograph is to provide a thorough study of supplements and projectivity conditions needed to investigate classes of modules related to lifting modules.The text begins with an introduction to small submodules, the radical, variations on projectivity, and hollow dimension. The subsequent chapters consider preradicals and torsion theories (in particular related to small modules), decompositions of modules (including the exchange property and local semi-T-nilpotency), supplements in modules (with specific emphasis on semilocal endomorphism rings), finishing with a long chapter on lifting modules, leading up their use in the theory of perfect rings, Harada rings, andquasi-Frobenius rings.Most of the material in the monograph appears in book form for the first time. The main text is augmented by a plentiful supply of exercises together with comments on further related material and on how the theory has evolved.

Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. This duality exhibits a certain asymmetry. While the theory of extending modules is well documented in monographs and text books, the purpose of this monograph is to provide a thorough study of supplements and projectivity conditions needed to investigate classes of modules related to lifting modules.

Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. This duality exhibits a certain asymmetry. While the theory of extending modules is well documented in monographs and text books, the purpose of this monograph is to provide a thorough study of supplements and projectivity conditions needed to investigate classes of modules related to lifting modules.

Preface		vii
Introduction		ix
Notation		xiii
Basic notions		1	(54)
Preliminaries		1	(10)
Small submodules and the radical		11	(9)
Cosmall inclusions and coclosed submodules		20	(6)
Projectivity conditions		26	(21)
Hollow dimension of modules		47	(8)
Preradicals and torsion theories		55	(48)
Preradicals and colocalisation		55	(15)
Torsion theories		70	(4)
Torsion theories related to small modules		74	(10)
Corational modules		84	(10)
Proper classes and τ-supplements		94	(9)
Decompositions of modules		103	(104)
The exchange property		103	(21)
LE-modules and local semi-T-nilpotency		124	(18)
Local direct summands		142	(20)
The total and LE-decompositions		162	(18)
Stable range 1 and cancellation		180	(14)
Decomposition uniqueness and biuniformity		194	(13)
Supplements in modules		207	(58)
Semilocal and weakly supplemented modules		207	(11)
Weak supplements and hollow dimension		218	(9)
Semilocal endomorphism rings		227	(6)
Supplemented modules		233	(24)
Submodules with unique coclosure		257	(8)
From lifting to perfect modules		265	(94)
Lifting modules		265	(12)
Finite direct sums of lifting modules		277	(12)
The lifting property for infinite sums		289	(7)
σ-lifting modules		296	(11)
Semi-discrete and quasi-discrete modules		307	(10)
Discrete and perfect modules		317	(14)
Injective modules lifting in σ[M]		331	(16)
Extending modules lifting in σ[M]		347	(12)
Appendix		359	(4)
Hall's Marriage Theorem		359	(3)
Konig's Graph Theorem		362	(1)
Bibliography		363	(24)
Index		387