Polynomial and power series rings. free algebras, firs and semifirs

Abstract

This chapter discusses polynomial and power series rings. It describes the weak algorithm, which provides a counterpart to the Euclidean algorithm, and it forms a natural tool for the study of polynomials in several noncommuting indeterminates. As in a Euclidean domain, every ideal is principal, so the (one-sided) ideals in a ring with weak algorithm are free, as modules over the ring, and this leads to firs and semifirs. Coproducts, a natural extension of free algebras, are also summarized. For any ring, R the polynomial ring R [ x ] is a familiar construction, obtained by adjoining to R an element x subject to the rule: ax = xa , for all a ∈ R . Moreover, every element of R [ x ] can be uniquely expressed as a polynomial in x with coefficients from R . In homological algebra, the global dimension of a ring forms a means of classification, and the hereditary rings—that is, the rings of global dimension 1—are simplest after the familiar case of global dimension 0 (the semisimple rings). A second mode of classification is based on the form taken by projective modules. The simplest class is formed by the projective-free rings, in which every finitely generated projective right module is free, of unique rank.