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How far is a Mori domain from being a Krull domain ? (English) Zbl 0623.13008
Let A be a Mori domain (so a.c.c. holds for integral divisorial ideals). Let \(D_ m(A)\) be the set of maximal integral divisorial ideals and let \({\mathcal I}(A)\) be all P in \(D_ m(A)\) satisfying the equivalent conditions: \((1)\quad A_ P\) is a DVR; \((2)\quad (P(A:P))_ v=A\) where \(X_ v=A:(A:X)\); \((3)\quad P:P\neq A:P.\) The condition \(P:P=A:P\) was studied by V. Barucci [J. Algebra 99, 132-142 (1986; Zbl 0596.13002)].
The main result is that \(A=B\cap A'\) where \(B=\cap \{A_ P| P\in {\mathcal I}(A)\}\quad is\) a Krull domain and \(A'=\cap \{A_ Q| Q\in D_ m(A)\setminus {\mathcal I}(A)\},\quad {\mathcal I}(A')\) being empty. Let \(C=\cap \{V_ j| j\in J\}\quad be\) a Krull domain where each \(V_ j\) is a DVR with maximal ideal \(M_ j\). For each j, let \(A_ j\) be the inverse image in \(V_ j\) of a subfield of \(V_ j/M_ j\). The domain \(\cap \{A_ j| j\in J\}\quad is\) Mori. Examples of this type, in particular where \(A_ j\neq V_ j\) for only one j, are studied.
Reviewer: C.P.L.Rhodes

MSC:
13E99 Chain conditions, finiteness conditions in commutative ring theory
13G05 Integral domains
13B30 Rings of fractions and localization for commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
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