Elementary Factorization Theory

Abstract

In this chapter we consider the problem of decomposing elements of a given commutative integral domain as products of irreducible elements. In a number of important integral domains such factorizations exist for all the non-units, and in a certain sense uniqueness of factorization holds. In these instances we can determine all of the factors of a given element and hence we can give simple conditions for the solvability of equations of the form ax = b. Since the factorization theory that we shall consider is a purely multiplicative theory that concerns the semi-group of non-zero elements of a commutative integral domain, we shall find it clearer to begin our discussion with the factorization theory of semi-groups.