Alexandroff Topology of Algebras over an Integral Domain

Abstract

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