LAUSR

Let S be an integral domain with field of fractions F 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 $$S {\setminus } \{ 0 \}$$ S \ { 0 } is A . Let $${\mathbb {S}}$$ S be the set of all S -nice subalgebras of A . We define a notion of open sets on $${\mathbb {S}}$$ S which makes this set a $$T_0$$ T 0 -Alexandroff space. This enables us to study the algebraic structure of $${\mathbb {S}}$$ S from the point of view of topology. We prove that an irreducible subset of $${\mathbb {S}}$$ S has a supremum with respect to the specialization order. We present equivalent conditions for an open set of $$\mathbb S$$ S to be irreducible, and characterize the irreducible components of $${\mathbb {S}}$$ S . We also characterize quasi-compactness of subsets of a $$T_0$$ T 0 -Alexandroff topological space.