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In this paper, we deal with the study of intermediate domains between a domain and a domain such that is an intersection of localizations of , namely the pair . More precisely, we study the pair and the pair , where and . We prove that, if is a Jaffard domain, then is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if is an -domain, then is a residually algebraic pair (that is for each intermediate domain between and , if is a prime ideal of ...
A domain R is called a maximal non-Jaffard subring of a field L if R ⊂ L, R is not a Jaffard domain and each domain T such that R ⊂ T ⊆ L is Jaffard. We show that maximal non-Jaffard subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dim R = dim R + 1. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when R is integrally...
We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards.
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