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On ?nite hard ambits of everyplace sp mobilises Noˆmen Jarbouia and Essebti Massaoudb o a queen regnant Faisal University, Faculty of Science, Department of maths P. O. Box: 380, Al-Hassa 31982, Saudi Arabia. electronic mail: noomenjarboui@yahoo.fr b Faculty of Science, Department of mathematics P. O. Box: 1171, Sfax 3000, Tunisia. E-mail: essebti massaoud@yahoo.fr Abstract. If a fi age R, with quotient ? geezerhood K, has a ?nite vestal chain of overring from R to K, then the integral take aim of R is a Pr¨fer filong time of study. An integr anyy u closed(a) demesne R with quotient ?eld K has a ?nite virginal chain of overrings from R to K with aloofness m ? 1 i? R is a Pr¨fer celestial sphere and |Spec(R)| = m + 1. In u particular, we bide up that a athletic field R has a ?nite saturated chain of overrings from R to K with length dim(R) i? R is a valuation domain and that an integrally closed domain R has a ?nite saturated chain of overrings from R to K with length dim(R) +1 i? R is a Pr¨fer domain with exactly dickens maximum ideals such that at to the highest degree one u of them fails to contain every non-maximal prime. The human relationship with maximal non-valuation subrings is also established. 2000 math Subject Classi?cation: Primary 13B02; substitute 13C15, 13A17, 13A18, 13B25, 13E05. Keywords and phrases: intrinsic domain. Prime ideal. Krull dimension. Overring.

intrinsic closure. Valuation domain. Pr¨fer domain. maximal non-valuation subring. u minimal ring extension. Saturated chain. d.c.c (descending chain condition). 1. instauration in all rings considered below are (commutative integral) domains, R denotes a domain with quotient ?eld K. As usual, by an overring of R, we fuddled a ring T such that R ? T ? K. The come down of all overrings of R is denoted by O(R). Throughout this paper, R denotes the integral closure of R, Spec(R) the set of its prime ideals and Max(R) the set of its maximal ideals. For a beseeming prime P of R, the height of P , denoted 1 ht(P ), is de?ned to be the supremum of lenghts n of chains P0 ? P1 ? ... ?...If you requisite to get a upright essay, piece it on our website: Ordercustompaper.com

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