# Article

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Keywords:
integral domain; intermediate ring; overring; integrally closed; Prüfer domain; residually algebraic pair; normal pair; primitive extension; a.c.c.; d.c.c.; minimal condition; maximal condition; affine extension; Dilworth number; width of an ordered set
Summary:
A ring extension $R\subseteq S$ is said to be FO if it has only finitely many intermediate rings. $R\subseteq S$ 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 $R\subseteq S$ 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 Prüfer and integral domains satisfying the corresponding finiteness conditions.
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