Arithmetical Characterizations of Divisor Class Groups Ii

Almost 20 years ago, W. Narkiewicz posed the problem to give an arithmetical characterization of the ideal class group of an algebraic number field ([13, problem 32]). In the meantime there are various answers to this question if the ideal class group has a special form. (cf. [4], [5], [12] and the literature cited there). The general case was treated by J. Koczorowski [11], F. Halter-Koch [8], [9, §5] and D. E. Rush [16]. In principle they proceed in the following way: they consider a finite sequence (ai)i=1...r of algebraic integers, requiring a condition of independence and a condition of maximality. Thereby the condition of independence guarantees that the ideal classes gi of one respectively all prime ideals gi appearing in the prime ideal decomposition of ai are independent in a group theoretical sense. The invariants of the class group are extracted from arithmetical properties of the ai’s, and the condition of maximality ensures that one arrives at the full class group but not at a subgroup. We study the problem in the general context of semigroups with divisor theory where every divisor class contains a prime divisor (cf. [1], [17]). Semigroups with divisor theory have turned out to be not only the appropriate setting for investigations on the arithmetic of rings of integers but to be of independent interest (cf. [6], [9], [10]). But contrary to the case of algebraic number fields, where the class group is always finite, every abelian group can be realized as a divisor class group of a semigroup with divisor theory ([17, Theorem 3.7] and [9, Satz 5]). The condition, that every divisor class has to contain at least one prime divisor, means a quite natural restriction. It is just this condition, which ensures that the relationship between the arithmetic of the semigroup and the class group in close enough, to allow a reasonable answer to the present problem. However, there are Dedekind domains which do not satisfy this condition, as can be seen from L. Skula’s paper [18]. We achieve the various descriptions of invariants of the class group by using only properties, which are satisfied by the semigroup if and only if they are satisfied by