Characterizing the ring extensions that satisfy FIP or FCP

Several parallel characterizations of the FIP and FCP properties are given. Also, a number of results about FCP are generalized from domains to arbitrary (commutative) rings. Let R⊆S be rings, with the integral closure of R in S. Then R⊆S satisfies FIP (resp., FCP) if and only if both and satisfy FIP (resp., FCP). If R is integrally closed in S, then R⊆S satisfies FIP ⇔ R⊆S satisfies FCP ⇔ (R,S) is a normal pair such that SuppR(S/R) is finite. If R⊆S is integral and has conductor C, then R⊆S satisfies FCP if and only if S is a finitely generated R-module such that R/C is an Artinian ring. The characterizations of FIP and FCP for integral extensions feature natural roles for the intermediate rings arising from seminormalization and t-closure.