FPF Ring Theory: Faithful Modules and Generators of Mod-R: 88 (London Mathematical Society Lecture Note Series, Series Number 88)
Cambridge University Press, 4/26/1984
EAN 9780521277389, ISBN10: 0521277388
Paperback, 176 pages, 23.1 x 15.2 x 1.3 cm
Language: English
This is the first book on the subject of FPF rings and the systematic use of the notion of the generator of the category mod-R of all right R-modules and its relationship to faithful modules. This carries out the program, explicit of inherent, in the work of G Azumaya, H. Bass, R. Dedekind, S. Endo, I. Kaplansky, K. Morita, T. Nakayama, R. Thrall, and more recently, W. Brandal, R. Pierce, T. Shores, R. and S. Wiegand and P. Vamos, among others. FPF rings include quasi-Frobenius rings (and thus finite rings over fields), pseudo-Frobenius (PF) rings (and thus injective cogenerator rings), bounded Dedekind prime rings and the following commutative rings; self-injective rings, Prufer rings, all rings over which every finitely generated module decomposes into a direct sum of cyclic modules (=FGC rings), and hence almost maximal valuation rings. Any product (finite or infinite) of commutative or self-basic PFP rings is FPF. A number of important classes of FPF rings are completely characterised including semiprime Neotherian, semiperfect Neotherian, perfect nonsingular prime, regular and self-injective rings. Finite group rings over PF or commutative injective rings are FPF. This work is the culmination of a decade of research and writing by the authors and includes all known theorems on the subject of noncommutative FPF rings. This book will be of interest to professional mathematicians, especially those with an interest in noncommutative ring theory and module theory.
Preface
Dedication and acknowledgement
Introduction
1. The basics
2. Noncommutative semiperfect and semiprime (C) FPF rings
3. Nonsingular FPF rings
4. Goldie prime FPF rings with RRM and the structure of neotherian prime FPF rings
5. Self-injective FPF rings, thin rings and FPF group rings
Summary of the structure of FPF rings
Open questions
Bibliography
Abbreviations and symbols
Index.