Czechoslovak Mathematical Journal, Vol. 70, No. 1, pp. 251-260, 2020
On strongly affine extensions of commutative rings
Received May 14, 2018. Published online November 18, 2019.
Abstract: A ring extension $R\subseteq S$ is said to be strongly affine if each $R$-subalgebra of $S$ is a finite-type $R$-algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R\subseteq S$ is integrally closed and strongly affine if and only if $R\subseteq S$ is a Prüfer extension (i.e. $(R,S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown. Let $G$ be a subgroup of the automorphism group of $S$ such that $R$ is invariant under action by $G$. If $R\subseteq S$ is strongly affine, then $R^G\subseteq S^G$ is strongly affine under some conditions.
Keywords: strongly affine; Prüfer extension; finitely many intermediate algebras property extension; finite chain propery extension; normal pair; integrally closed pair; ring of invariants