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Commutative rings --- Semigroup rings --- 512.55 --- 512.55 Rings and modules --- Rings and modules --- Rings, Semi group --- Rings, Semigroup --- Semi group rings --- Rings (Algebra) --- Semigroups --- Ordered algebraic structures

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Problem-solving is an art central to understanding and ability in mathematics. With this series of books, the authors have provided a selection of worked examples, problems with complete solutions and test papers designed to be used with or instead of standard textbooks on algebra. For the convenience of the reader, a key explaining how the present books may be used in conjunction with some of the major textbooks is included. Each volume is divided into sections that begin with some notes on notation and prerequisites. The majority of the material is aimed at the students of average ability but some sections contain more challenging problems. By working through the books, the student will gain a deeper understanding of the fundamental concepts involved, and practice in the formulation, and so solution, of other problems. Books later in the series cover material at a more advanced level than the earlier titles, although each is, within its own limits, self-contained.

Group algebras --- Algebraic fields --- Rings (Algebra)

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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.

FPF rings. --- Associative rings. --- Modules (Algebra) --- Categories (Mathematics) --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Finitely pseudo-Frobenius rings --- Rings, Finitely pseudo-Frobenius --- Rings, FPF --- Associative rings --- Algebres et anneaux associatifs --- Ideaux et modules

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512.7 --- 512.7 Algebraic geometry. Commutative rings and algebras --- Algebraic geometry. Commutative rings and algebras --- Algebra

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This book is a self-contained account of the theory of classgroups of group rings. The guiding philosophy has been to describe all the basic properties of such classgroups in terms of character functions. This point of view is due to A. Frohlich and it achieves a considerable simplification and clarity over previous techniques. A main feature of the book is the introduction of the author's group logarithm, with numerous examples of its application. The main results dealt with are: Ullom's conjecture for Swan modules of p-groups; the self-duality theorem for rings of integers of tame extensions; the fixed-point theorem for determinants of group rings; the existence of Adams operations on classgroups. In addition, the author includes a number of calculations of classgroups of specific families of groups such as generalized dihedral groups, and quaternion and dihedral 2-groups. The work contained in this book should be readily accessible to any graduate student in pure mathematics who has taken a course in the representation theory of finite groups. It will also be of interest to number theorists and algebraic topologists.

Group rings. --- Class groups (Mathematics) --- Modules (Algebra) --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Groups, Class (Mathematics) --- Algebraic number theory --- Commutative rings --- Ideals (Algebra) --- Group theory --- Ordered algebraic structures --- Class groups (Mathematics). --- Modules (Algebra). --- Anneaux (algèbre)