ABSTRACT
CHAPTER 2
A polynomial ring in many variables over a field is factorial and thus a very simple object in factorization theory, but it has a very complicated ideal theory. This simple example already indicates that in factorization theory, the theory of v-ideals (or divisorial ideals) as introduced by H. Pru¨fer [341] and W. Krull (see [288], [289], [290]) is a more suitable tool than Dedekind’s ideal theory. According to I. Arnold [31] and P. Lorenzen [306], v-ideals are purely multiplicative and can be developed in the framework of monoids. For an integral domain R, v-ideals of R are essentially the same as v-ideals of the monoid R• (see Section 2.10). In this volume we describe the arithmetic of a domain R by means of the monoid R•. For this purpose the theory of v-ideals in monoids provides the appropriate ideal-theoretic basis.
