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Globalizing local properties of Prüfer domains. (English) Zbl 0928.13013
A property \({\mathcal P}\) is called a globalizing property for a class \({\mathcal C}\) of integral domains if the property \({\mathcal P}\) holds for each localization \(R_M\) of an \(R\) in \({\mathcal C}\) at each maximal ideal \(M\) of \(R\). In this paper, the author studies several globalizing properties for different classes of Prüfer domains with particular emphasis on when local divisoriality and invertibility properties can be globalized. Some of the properties and classes of Prüfer domains studied include \(SV\)-stable domains, \(h\)-local domains, divisorial and totally divisorial domains, property \((\# \#)\), and strongly discrete Prüfer domains.

MSC:
13G05 Integral domains
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13B30 Rings of fractions and localization for commutative rings
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[1] Anderson, D.D.; Huckaba, J.A.; Papick, I.J., A note on stable domains, Houston J. math., 13, 13-17, (1987) · Zbl 0624.13002
[2] Bass, H., On the ubiquity of Gorenstein rings, Math. Z., 82, 8-28, (1963) · Zbl 0112.26604
[3] Bazzoni, S.; Salce, L., Warfield domains, J. algebra, 185, 836-868, (1996) · Zbl 0873.13020
[4] Facchini, A., Generalized Dedekind domains and their injective modules, J. pure appl. algebra, 94, 159-173, (1994) · Zbl 0845.13005
[5] Fontana, M.; Huckaba, J.; Papick, I., Some properties of divisorial prime ideals in Prüfer domains, J. pure appl. algebra, 39, 95-103, (1986) · Zbl 0578.13002
[6] Fontana, M.; Huckaba, J.; Papick, I., Prüfer domains, (1997), Dekker New York
[7] Fontana, M.; Huckaba, J.; Papick, I.; Roitman, M., Prüfer domains and endomorphism rings of their ideals, J. algebra, 157, 489-516, (1993) · Zbl 0779.13008
[8] Fontana, M.; Popescu, N., Sur une classe d’anneaux généralisent LES anneaux de Dedekind, J. algebra, 173, 44-66, (1995) · Zbl 0844.13012
[9] S. Gabelli, A class of Prüfer domains with nice divisorial ideals, in, Proceedings on the Second International Conference on Commutative Ring Theory, Fès, 1995, Lecture Notes in Pure and Applied Mathematics, Dekker, New York
[10] S. Gabelli, N. Popescu, Invertible and divisorial ideals of generalized Dedekind domains, J. Pure Appl. Algebra · Zbl 0931.13010
[11] Gilmer, R., Overrings of Prüfer domains, J. algebra, 4, 331-340, (1966) · Zbl 0146.26205
[12] Gilmer, R.; Heinzer, W., Overrings of Prüfer domains, II, J. algebra, 7, 281-302, (1967) · Zbl 0156.04304
[13] Heinzer, W., Integral domains in which each non-zero ideal is divisorial, Mathematika, 15, 164-170, (1979) · Zbl 0169.05404
[14] Heinzer, W.; Papick, I., The radical trace property, J. algebra, 112, 110-121, (1988) · Zbl 0641.13001
[15] Huckaba, J.; Papick, I., When the dual of an ideal is a ring, Manuscripta math., 37, 67-85, (1982) · Zbl 0484.13001
[16] Krull, W., Idealtheorie, (1935), Springer-Verlag Berlin
[17] Larsen, M.; McCarthy, P., Multiplicative theory of ideals, (1971), Academic Press New York · Zbl 0237.13002
[18] Lucas, T., The radical trace property and primary ideals, J. algebra, 184, 1093-1112, (1996) · Zbl 0868.13016
[19] Matlis, E., Torsion-free modules, (1972), Univ. of Chicago Press Chicago · Zbl 0298.13001
[20] Ohm, J.; Pendleton, R., Rings with Noetherian spectrum, Duke math. J., 35, 631-639, (1968) · Zbl 0172.32201
[21] Popescu, N., A class of Prüfer domains, Rev. roumaine math. pures appl., 29, 777-786, (1984) · Zbl 0564.13011
[22] Sally, J.; Vasconcelos, W., Stable rings, J. pure appl. algebra, 4, 319-336, (1979) · Zbl 0284.13010
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