Fundamental Concepts in the Theory of Algebras

Overview

Let R be a commutative ring. This first chapter recalls some fundamental concepts and constructions from algebra, e.g., involutions and gradings on R-algebras, and tensor products and graded tensor products of R-algebras. These are also the general Leitmotifs of this book. In addition, this chapter introduces free quadratic R-algebras, i.e., algebras of the form R[X]/(X² - aX - b), and some of their basic properties. These algebras provide concrete illustrations of the basic concepts and constructions just referred to; more significantly, they will be of importance in virtually every chapter of this book.