Characterizing the ring extensions that satisfy FIP or FCP
Let \(R\) and \(S\) be two commutative rings. The extension \(R \subseteq S \) is said to satisfy FIP (for the ``finitely many intermediate algebras property'') if the set of all \(R\)-subalgebras of \(S\) is finite. The extension \(R \subseteq S \) is said to satisfy FCP (for the ``finite chain property'') if each chain of \(R\)-subalgebras of \(S\) is finite. It is clear that each extension that satisfies FIP must also satisfy FCP; the converse is false, as can be easily seen via field-theoretic examples. In the present paper, the authors prove several (parallel) characterizations of the FIP and FCP properties. Note that a number of results about FCP are generalized from domains to arbitrary (commutative) rings. Among the main results of the paper, I would like to mention the following. (a) Let \(R \subseteq S \) be an extension of rings rings, with \(\overline R\) the integral closure of \(R \) in \(S\). Then \(R \subseteq S \) satisfies FIP (respectively, FCP) if and only if both \(R \subseteq \overline R \) and \(\overline R \subseteq S \) satisfy FIP (respectively, FCP). (b) If \(R\) is integrally closed in \(S\), then \(R \subseteq S \) satisfies FIP if and only if \(R \subseteq S \) satisfies FCP if and only if \((R, S)\) is a normal pair such that Supp\(_R (S/R)\) is finite. (c) If \(R \subseteq S \) is integral and has conductor \(C\), then \(R \subseteq S \) satisfies FCP if and only if \(S\) is a finitely generated \(R\)-module such that \(R/C\) is an Artinian ring. Among the motivations of their study, the authors mention the fact that the characterizations of FIP and FCP for integral extensions feature natural roles for the intermediate rings arising from seminormalization and t-closure.
scientific article; zbMATH DE number 6166665
scientific article