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 R ¯ the integral closure of R in S . Then R ⊆ S satisfies FIP (resp., FCP) if and only if both R ⊆ R ¯ and R ¯ ⊆ S 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 Supp R ( 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.