On Finite Saturated Chains of Overrings |
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Authors: | Noômen Jarboui Essebti Massaoud |
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Institution: | 1. Faculty of Science, Department of Mathematics , King Faisal University , Al-Hassa , Saudi Arabia noomenjarboui@yahoo.fr;3. Faculty of Science, Department of Mathematics , Sfax , Tunisia |
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Abstract: | If a domain R, with quotient field K, has a finite saturated chain of overrings from R to K, then the integral closure of R is a Prüfer domain. An integrally closed domain R with quotient field K has a finite saturated chain of overrings from R to K with length m ≥ 1 iff R is a Prüfer domain and |Spec(R)| =m + 1. In particular, we prove that a domain R has a finite saturated chain of overrings from R to K with length dim(R) iff R is a valuation domain and that an integrally closed domain R has a finite saturated chain of overrings from R to K with length dim (R) +1 iff R is a Prüfer domain with exactly two maximal ideals such that at most one of them fails to contain every non-maximal prime. The relationship with maximal non-valuation subrings is also established. |
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Keywords: | (Descending chain condition) d c c Integral closure Integral domain Krull dimension Maximal non-valuation subring Minimal ring extension Overring Prime ideal Prüfer domain Saturated chain Valuation domain |
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