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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
##### Keywords:
Mori domain; a.c.c.; integral divisorial ideals; Krull domain
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##### References:
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