On the class group of a Mori domain

Abstract

It is well known that the monoid D(A) of the divisorial ideals of an integral domain A is a group if and only if A is completely integrally closed. In this case, one may also define the class group of A, C(A) = D(A)/P(A), where P(A) is the subgoup of principal ideals of A. The group C(A) sometimes gives good information about A; for example, if A is a Krull domain, C(A) = 0 if and only if A is a unique factorization domain [9, Proposition 6.11. In [6, 71, the class group C(A) is defined also for a noncompletely integrally closed domain A as C(A) = T(A)/P(A), where 7’(A) is the group of t-invertible t-ideals of A (see the definition in Sect. 1). The aim of this paper is to study the r-invertible t-ideals and the class group of a Mori domain. We recall that a Mori domain is a domain such that the ascending chain condition holds in the set of integral divisorial ideals. Noetherian domains are Mori domains and a Mori domain is a Krull domain if and only if it is completely integrally closed [9, Sect. 33. From this point of view, we begin by observing that in a Mori domain A the t-ideals are exactly the divisorial ideals (Proposition (1.1)); so that T(A) is the group of the invertible elements of D(A), i.e., the u-invertible ideals of A. Since in a Mori domain A a v-invertible divisorial prime is maximal divisorial (Proposition (1.3)), a complete characterization of u-invertible divisorial primes is found early on: they are the maximal divisorial primes P such that A, is a DVR (Corollary (1.4)). Also, we can give a characterization of a certain class of u-invertible divisorial ideals in terms of these primes, generalizing some well-known results for Krull domains