## Coherent Mori domains and the principal ideal theorem.(English)Zbl 0622.13007

Most of this paper concerns domains satisfying PIT, i.e. the conclusion of the principal ideal theorem. New classes of such domains are found, e.g. Mori such that divisorial primes P with $$ht(P)>1$$ are f.g. All overrings (in the quotient field) of a domain R satisfy PIT if and only if they have $$\cap I^ n=0\quad$$ for all proper principal ideals I. Call a domain quasicoherent (resp. finite conductor) if intersections of finitely many (resp. two) principal ideals are f.g. Our domain R is Noetherian if and only if $$ht(P)<\infty$$ and R/P is quasicoherent Mori for all primes P. For finite conductor domains, Mori implies PIT, integrally closed Mori is equivalent to Krull, and Mori with proper overrings satisfying PIT implies 1-dimensional Noetherian. Many criteria are given for a PIT domain to be 1-dimensional, e.g. Spec being a tree. For domains of global dimension 2 it is found that Mori is equivalent to Noetherian. Discussion of the stability of PIT embraces polynomial extensions of R, especially when $$\dim (R)=1$$. If R[X] satisfies PIT then so does R and R is an S-domain in the sense of I. Kaplansky [”Commutative rings” (2nd edition 1974; Zbl 0296.13001)].
Reviewer: C.P.L.Rhodes

### MSC:

 13E99 Chain conditions, finiteness conditions in commutative ring theory 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations

Zbl 0296.13001
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