Finitely star regular domains

Let R be an integral domain, Star(R) the set of all star operations on and StarFC(R) finite type R. Then is said to regular if |Star(T)|≤|Star(R)| for every overring T In this paper we introduce notion finitely domain as such that |StarFC(T)|≤|StarFC(R)| each First, show notions domains are completely different do not imply other. Next, extend/generalize well-known results regularity in Noetherian Prüfer contexts regularity. Also handle issued from classical pullback constructions construct regular.