Some finiteness chain conditions on the set of intermediate rings
Let \(R \subset S\) be an extension of integral domains, \([R,S]\) be the set of rings between \(R\) and \(S\), and \(K\) be the quotient field of \(R\). Let \(\mathrm{Supp}(S/R) = \{P \in\) Spec\((R)| PS = S\}\) and \(\overline{R_S}\) be the integral closure of \(R\) in \(S\). We say that the extension \(R \subset S\) satisfies the finite chain condition (FCC) if each chain of distinct rings between \(R\) and \(S\) is finite. It is known that \(R \subset K\) satisfies FCC (i.e., \(R\) is an FC-domain) if and only if \(\overline{R}\), the integral closure of \(R\) in \(K\), is a Prüfer domain with finite spectrum, (ii) \(\overline{R}\) is a finite \(R\)-module, and (iii) \(R/(R:\overline{R})\) is an Artinian ring where \((R:\overline{R}) = \{x \in R| x\overline{R} \subseteq R\}\) [\textit{R. Gilmer}, Proc. Am. Math. Soc. 131, No. 8, 2337--2346 (2003; Zbl 1017.13009)]. The author of the paper under review noted that (i) if \(R = K\), then \(R \subset S\) satisfies FCC if and only if \(R \subset S\) is an extension of fields with finite degree and (ii) if \(R \neq K\) and \(R \subset S\) satisfies FCC, then \(R \subset S \subseteq K\). Let \(S \subseteq K\). The main result proved in this paper is that \(R \subset S\) satisfies FCC if and only if there is a finite maximal chain \(R = R_0 \subset R_1 \subset \cdots \subset R_n = S\) in \([R,S]\) such that one of the \(R_i\) is \(\overline{R_S}\), if and only if (i) \((\overline{R_S}, S)\) is a normal pair (i.e., each \(T \in [\overline{R_S}, S]\) is integrally closed in \(S\)) and \(\mathrm{Supp}(S/R)\) is finite, (ii) \(\overline{R_S}\) is a finite \(R\)-module, and (iii) \((R:\overline{R_S})\) is an intersection of finitely many maximal ideals of \(R\).
scientific article; zbMATH DE number 5718200
scientific article