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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

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
Full Text: DOI
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