How far is a Mori domain from being a Krull domain?

Abstract

A Mori domain is a domain such that the ascending chain condition holds in the set of integral divisorial ideals (cf. [13, p. 195]). Thus in a Mori domain A we can consider the proper integral ideals maximal with respect to being divisorial. We call them maximal divisorial ideals of A and we denote this set by DIn(A). The ideals of DIn(A) are primes (cf. Proposition 2.1) and have a central role in this paper. We first generalize to any Mori domain some results known for Krull domains, which are actually completely integrally closed Mori domains (cf. for example [4, Ch. 7, §1, No. 3, Theorem 2]). It is well known that, i rA is a Krull domain, Dm(A ) is the set of height 1 primes and A = ~ {Ap: PeDro(A)}, where this decomposition has finiteness character and, for each PeDro(A), Ap is a DVR (discrete valuation ring). Moreover in a Krull domain A the group of the divisorial ideals is generated by the height 1 primes ofA (cf. [4, Ch. 7, §1]). We show that, i fA is a Mori domain, A = ~ {Ap: PeDro(A)}, where this decomposition has finiteness character (cf. Proposition 2.2(b)) and it is irredundant (cf. Corollary 3.2), and we give a 'good representation' for any divisorial ideal of A in terms of DIn(A) (cf. Proposition 2.2(c)). In order to 'measure' the distance that a Mori domain A is from being Krull, it seems natural to divide Dm(A) into two subsets: the primes P of Dm(A ) such that A e is a DVR, and the other primes. We characterize the first subset of Dm(A) (denoted by ,~(A)) in Theorem 2.5. In this way we get for a Mori domain A a 'canonical decomposition' in two overrings, A =B CIA', where B = ~ {Ap: P e ~(A)} is a Krull domain and A'= ~ {Ap: PeDro(A) ~(A)} is a Mori domain very far from being Krull (we call it a strongly Mori domain) (cf. Theorem 3.3). Therefore,