Set relations and set systems induced by some families of integral domains

In this paper, given an integral domain U, we investigate the main properties of a relation ←mod which is based on the interrelation between subdomains of U and finitely generated unitary submodules of U. We shall characterize it in terms of a second relation ≺◇ between n-tuples (u1,...,un) of elements of U and subdomains D of U defined by the vanishing in (u1,...,un) of some polynomial p(Z1,...,Zn) belonging to a specific subset of the polynomial ring in several variables D[Z1 , . . . , Zn ]. Such an equivalence shall be used in order to introduce three specific collections of subdomains XU , BU and PU , whose algebraic properties present a close connection with geometrical and combinatorial properties induced by ←mod. On the other hand, the characterization of the subdomains of XU leads to the more general problem of finding a map Ψ associating with a subdomain D of U a collection Ψ(D) of subdomains of KU such that the intersection of some or of any member of Ψ(D) gives D. In this perspective, in the present paper we shall study two further collections of subdomains of U,denotedrespectivelybyEU andLU,whosemainproperties are related to those of the families PU and BU . Finally, our investigation of all the aforementioned subdomain families shall be also related to the study of pairs (e, ξ), where e ∈ U ∖ {0} and ξ is an idempotent ring endomorphism of U whose kernel agrees with the ideal of U generated by e.