Ring extensions with some finiteness conditions on the set of intermediate rings

Abstract

A ring extension R ⊆ S is said to be FO if it has only finitely many intermediate rings. R ⊆ S is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension R ⊆ S to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair with only finitely many intermediate rings. We also obtain as a corollary several new and old characterizations of Prufer and integral domains satisfying the corresponding finiteness conditions.