IDEALS AND OVERRINGS OF DIVIDED DOMAINS

Year 2010, Volume: 8 Issue: 8, 80 - 113, 01.12.2010

Gabriel Picavet

Abstract

New properties of divided domains R are established by looking at multiplicatively closed subsets associated to ring morphisms. Let I be an ideal of R. We exhibit primary ideals, like I√I and In if I is primary. We show that Ass(I) = V(I) ∩ Spec(RMax(Ass(I))). Moreover, the image of the maximal spectrum of (I : I) is contained in Ass(I). We show that certain intersections of ideals are primary ideals. Goldman prime ideals are prime gideals. The characterization of maximal flat epimorphic subextensions gives as a result that R is a valuation subring of Pr¨ufer hulls. We characterize Fontana-Houston divided Ω-domains, divided APVDs and divided PPC-domains.

Keywords

affine open subset, almost pseudo-valuation domain, antesharp prime ideal, complete integral closure, conductor overring

References

• A. Badawi, On divided commutative rings, Comm. Algebra, 27(3) (1999), 1465–1474.
• A. Badawi and D. E. Dobbs, Some examples of locally divided rings, 73–83,
• Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001.
• A. Badawi and E. Houston, Powerful ideals, strongly primary ideals, almost
• pseudo-valuation domains, and conducive domains, Comm. Algebra, 30(4) (2002), 1591–1606.
• N. Bourbaki, Alg`ebre Commutative, Chapitre 4, Hermann, Paris, 1967.
• J. Coykendall and D. Dobbs, Fragmented ideals have infinite Krull dimension,
• Rend. Circ. Mat. Palermo, 50(2) (2001), 377-388.
• G. H. Chang, H. Nam and J. Park, Strongly primary ideals, 378–386, Lecture
• Notes Pure Appl. Math., 241, Chapman & Hall, Boca Raton Fl, 2005.
• D. E. Dobbs, Divided rings and going-down, Pacific J. Math., 67 (1976), 353–
• D. E. Dobbs, On going-down for simple overrings, III, Proc. Amer. Math.
• Soc., 54 (1976), 35–38.
• D. E. Dobbs, Coherence, ascent of going-down and pseudo-valuation domains,
• Houston J. Math., 4 (1978), 551–567.
• D. E. Dobbs, Ahmes expansions of formal Laurent series and a class of NonArchimedean domains, J. Algebra, 103 (1986), 193–201.
• D. E. Dobbs, Fragmented integral domains, Portugaliae Math., 45 (1985– 1986), 463–473.
• D. E. Dobbs, On flat divided prime ideals, 305–315, Factorization in integral domains, Lecture Notes Pure Appl. Math., 189, Dekker, New York, 1997.
• D. E. Dobbs, Recent progress on going-down I, Non-Noetherian commutative ring theory, 139–168, Kluwer Acad Publ., Dordrecht, 2000.
• D. E. Dobbs and R. Fedder, Conducive integral domains, J. Algebra, 86 (1984), 494–510.
• D. E. Dobbs, R. Fedder and M. Fontana, G-domains and spectral spaces, J. Pure Appl. Algebra, 51 (1988), 89–110.
• D. E. Dobbs and G. Picavet, Straight rings, Comm. Algebra, 37(3) (2009), 757–793.
• D. E. Dobbs and G. Picavet, Straight rings, II, Commutative Algebra and its Applications, 183–205, de Gruyter, Berlin, New York, 2009.
• C. Faith, Annihilator ideals, associated primes and Kasch-McCoy commuta- tive rings, Comm. Algebra, 19(7) (1991), 1867–1892.
• C. Faith, Rings with few zero divisors are those with semilocal Kasch quotient rings, Houston J. Math., 22(4) (1996), 687–690.
• M. Fontana, Kaplansky ideal transform: A survey, 271–306, Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999.
• M. Fontana and E. Houston, On integral domains whose overrings are Ka- plansky ideal transforms, J. Pure Appl. Algebra, 163 (2001), 173–192.
• M. Fontana, E. Houston and T. G. Lucas, Toward a classification of prime ideals in Pr¨ufer domains, Forum Math., 19 (2007), 971–1004.
• M. Fontana, J. A. Huckaba and I. J. Papick, Pr¨ufer Domains, Dekker, New York, 1997.
• M. Fontana, J. A. Huckaba, I. J. Papick and M. Roitman, Pr¨ufer domains and endomorphism rings of their ideals, J. Algebra, 157 (1993), 489–516.
• M. Fontana and N. Popescu, Universal property of the Kaplansky ideal trans- form and affineness of open subsets, J. Pure Appl. Algebra, 173 (2002), 121–
• R. M. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973.
• L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without finiteness conditions, primal ideals, Trans. Amer. Math. Soc., 357 (2004), 2771–2798.
• L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without finiteness conditions: Irreducibility in the quotient field, 121–145, Lect. Notes Pure Appl. Math., 249, Chapman & Hall, Boca Raton, FL, 2006. 215–239.
• L. Fuchs and L. Salce, Modules over Non-Noetherian Domains, Math. Surveys and Monographs, 84, American Math. Society, 2001.
• R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math., 90, 1992.
• R. Gilmer and W. Heinzer, Imbeddability of a commutative ring in a finite dimensional ring, Manuscripta Math., 84 (1994), 401–414.
• A. Grothendieck and J. Dieudonn´e, El´ements de G´eom´etrie Alg´ebrique, Springer Verlag, Berlin, 1971.
• J. A. Huckaba and I. J. Papick, When the dual of an ideal is a ring, Manuscripta Math., 37 (1982), 67–85.
• M. Kanemitsu, R. Matsuda, N. Onoda and T. Sugatani, Idealizers, complete integral closures and almost pseudo-valuation domains, Kyungpook Math. J., 44 (2004), 557–563.
• I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
• M. Knebusch and D. Zhang, Manis valuations and Pr¨ufer extensions, Lecture Notes in Mathematics, Vol. 1791, Springer Verlag, Berlin, 2002.
• R. A Kuntz, Associated prime divisors in the sense of Krull, Canad. J. Math., 24 (1972), 808–818.
• D. Lazard, Autour de la platitude, Bull. Soc. Math. France, 97 (1969), 81–128.
• K. Morita, Flat modules, injective modules and quotient rings, Math. Z., 120 (1971), 25–40.
• A. Okabe, On conductor overrings of an integral domain, Tsukuba J. Math., 8 (1984), 69–75.
• A. Okabe, Some ideal-theoretical characterizations of divided domains, Hous- ton J. Math., 12(4) (1986), 563–577.
• I. Papick, Topologically defined classes of going-down domains, Trans. Amer. Math. Soc., 219 (1976), 1–37.
• G. Picavet, Sur une g´en´eralisation de la notion de spectre d’anneaux, Ann. Sci. Univ. Clermont, 44 (1970), 81–101.
• G. Picavet, Sur les anneaux commutatifs dont tout ideal premier est de Gold- man, C. R. Acad. Sci. Paris, Ser. A, 280 (1975), 1719–1721.
• G. Picavet, Propri´et´es et applications de la notion de contenu, Comm. Alge- bra, 13(10) (1985), 2231–2265.
• G. Picavet, Puret´e, rigidit´e et morphismes entiers, Trans. Amer. Math. Soc., 323 (1991), 283–313.
• G. Picavet, Geometric subsets of a spectrum, 387–417, Lecture Notes in Pure and Appl. Math., 231, Dekker, New York, 2003.
• G. Picavet, Treed domains, Int. Electron. J. Algebra, 3 (2008), 1–14.
• R. G. Swan, On seminormality, J. Algebra, 67 (1980), 210–229.
• S. Singh and P. Manchand, On complete integral closure of G-domains, Indian J. Pure Appl. Math., 20 (1989), 884–886. Gabriel Picavet
• Laboratoire de Math´ematiques
• Universit´e Blaise Pascal 63177 Aubiere Cedex
• e-mail: [e-mail]