Transfer Results for the FIP and FCP Properties of Ring Extensions

Abstract

For an extension E: R ⊂ S of (commutative) rings and the induced extension F: R(X) ⊂ S(X) of Nagata rings, the transfer of the FCP and FIP properties between E and F is studied. Then F has FCP ⇔ E has FCP. The extensions E for which F has FIP are characterized. While E has FIP whenever F has FIP, the converse fails for certain subintegral extensions; it does hold if E is integrally closed, seminormal, or subintegral with R quasi-local having infinite residue field. If F has FIP, conditions are given for the sets of intermediate rings of E and F to be order-isomorphic.