Lifting Modules

by Clark, John

• ISBN13: 9783764375720
• ISBN10: 3764375728
• Format: Paperback
• Copyright: 2006-09-30
• Publisher: Birkhauser

Summary

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, and quasi-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.

Table of Contents

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