# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Arnold, J.T.; Brewer, J.W., On flat overrings, ideal transforms and generalized transforms of a commutative ring, J. algebra, 18, 254-263, (1971) · Zbl 0218.13019 [2] Barucci, V., On a class of Mori domains, Comm. algebra, 11, 1989-2001, (1983) · Zbl 0518.13012 [3] Barucci, V., Strongly divisorial ideals and complete integral closure of an integral domain, J. algebra, 99, 132-142, (1986) · Zbl 0596.13002 [4] Bourbaki, N., Algèbre commutative, (1961-1965), Hermann Paris, Chapters 1-7 · Zbl 0141.03501 [5] Fontana, M., Topologically defined classes of commutative rings, Ann. mat. pura appl. (4), 123, 331-355, (1980) · Zbl 0443.13001 [6] Fossum, R., The divisor class group of a Krull domain, (1973), Springer Berlin · Zbl 0256.13001 [7] Gilmer, R., Multiplicative ideal theory, (1972), Marcel Dekker New York · Zbl 0248.13001 [8] Heinzer, W.; Ohm, J., Noetherian intersections of integral domains, Trans. amer. math. soc., 167, 291-308, (1972) · Zbl 0239.13013 [9] Heinzer, W.; Ohm, J.; Pendleton, R.L., On integral domains of the form ∩DP, P minimal, J. reine angew. math., 241, 147-159, (1970) · Zbl 0191.32202 [10] Kaplansky, J., Commutative rings, (1970), Allyn and Bacon Boston · Zbl 0203.34601 [11] Querré, J., Sur une propriété des anneaux de Krull, Bull. sci. math., 95, 341-354, (1971) · Zbl 0219.13015 [12] Querré, J., Sur LES anneaux réflexifs, Canad. J. math., 6, 1222-1228, (1975) · Zbl 0335.13010 [13] Querré, J., Cours d’ algèbre, (1976), Masson Paris [14] Raillard, N., Sur LES anneaux de Mori, C.R. acad. sci. Paris, 280, 1571-1573, (1975) · Zbl 0307.13010 [15] Raillard, N., Sur LES anneaux de Mori, These, (1976), Paris VI · Zbl 0374.13013 [16] Zafrullah, M., The v-operation and intersections of quotient rings of integral domains, Comm. algebra, 13, 1699-1712, (1985) · Zbl 0573.13002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.