Algebra, Volume 2: Rings

This is the first volume of the book Algebra planned by the authors to provide adequate preparation in algebra to prospective teachers and researchers in mathematics and related areas. Beginning with groups of symmetries of plane configurations, it studies groups (with operators) and their homomorphisms, presentations of groups by generators and relations, direct and semidirect products, Sylow's theorems, soluble, nilpotent and Abelian groups. The volume ends with Jordan's classification of finite subgroups of the group of orthogonal transformations of R3. An attractive feature of the book is its richness in purposeful examples and instructive exercises with a focus on the roots of algebra in number theory, geometry and theory of equations Aiming to provide adequate preparation in algebra to prospective teachers and researchers in mathematics and related areas, this book delves into the intricacies of Algebra. Starting with the groups of symmetries of plane configurations, then moving onto study groups (with operators) and their homomorphisms, before diving into the complexities of Sylow's theorems and Abelian groups. All of these topics and many others are richly supported by instructive exercises and purposeful examples, with a focus on the roots of algebra in geometry and number theory. The volume ends with Jordan's classification of finite subgroups of the group of orthogonal transformations of R3. Table of Contents Preface Notations Preliminaries Groups of Plane Symmetries Groups and Their Homomorphisms Subgroups, Normal Subgroups and Quotient Groups Homomorphism Theorems Direct and Semidirect Products Finite Groups Series of Subgroups Abelian Groups Finite Subgroups of Orthogonal Groups Bibliography Index Continuing their exposition of algebra begun in Volume 1, in this book the authors present the complexities associated with the study of groups. A Plethora of topics are explored such as integral and algebraic elements and the rings of endomorphisms and fractions and ideal theory, all of which are supported by illustrated exercises and examples.