Domain theory aspects of extensions of integral domains

Let S be an integral domain which is not a field, let F be its field of fractions, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R is lying over S and the localization of R with respect to S∖{0} is A. Let S be the set of all S-nice subalgebras of A. In [14] we showed that S can be given the structure of a T₀ Alexandroff topological space. In this paper we show that under the specialization order S becomes a dcpo, and we study it from the additional point of view of domain theory. We prove that sup[Formula presented]supD whenever D is a directed subset of S such that [Formula presented]. We also present equivalent conditions for S to be a continuous dcpo. Assuming A is finitely generated as an algebra over F, we prove that the compact elements of S are precisely the S-nice subalgebras of A that are finitely generated as an algebra over S. We then prove that S is an algebraic domain, and discuss the notion of immediate neighbors in S. Under the additional assumption that A contains an S-stable basis, we show that there exists an infinite strictly decreasing chain of infinite ideals of S. At the end, we present two insightful constructions in which A is not finitely generated as an algebra over F, and S does not have a basis.