Ideals and Overrings of Divided Domains
International Electronic Journal of Algebra Volume 8 (2009) 80-113 IDEALS AND OVERRINGS OF DIVIDED DOMAINS Gabriel Picavet Received: 27 November 2009; Revised: 3 June 2010 Communicated by Abdullah Harmanc³ Abstract. New properties of divided domains R are established by looking at multiplicatively closed subsets associated to ring morphisms. Let I be an p 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 g- ideals. The characterization of maximal flat epimorphic subextensions gives as a result that R is a valuation subring of PrÄuferhulls. We characterize Fontana- Houston divided -domains, divided APVDs and divided PPC-domains. Mathematics Subject Classi¯cation (2000): Primary 13G05; Secondary 13A15, 13B24, 13F05 Keywords: a±ne open subset, almost pseudo-valuation domain, antesharp prime ideal, complete integral closure, conductor overring, divided domain (ring), (flat) epimorphism, fragmented domain, g-ideal ring, G-ideal ring, going-down domain, i-domain, Kaplansky transform, Kasch ring, Manis pair, maximal flat epimorphic (sub)extension, -domain, open domain, PPC-domain, power-Ahmes domain, propen domain, primal ideal, primary ideal, treed do- main, unbranched prime ideal, valuation pair. 1. Introduction and notation This paper deals with commutative unital rings and their (homo)morphisms. Dobbs introduced the divided property [7]. Let R be an integral domain with quotient ¯eld K, with R 6= K (i.e.
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