On finiteness of chains of intermediate rings
Let \(R\subseteq S\) be an extension of integral domains. This extension is said to have the ``finite length of intermediate chains of domains'' property (FICP) if each chain of intermediate rings between \(R\) and \(S\) is finite. Let \(T\) be a ring between \(R\) and \(S\). Certainly if \(R\subseteq S\) has FICP, so do the extensions \(R\subseteq T\) and \(T\subseteq S\). The converse is false as \( \mathbb{Z}\subseteq \mathbb{Z}_{(p)}\) and \(\mathbb{Z}_{(p)}\subseteq \mathbb{ Q}\) both have FICP for any nonzero prime \(p\), but \(\mathbb{Z}\subseteq \mathbb{Q}\) certainly does not. Now \textit{R. Gilmer} [Proc. Am. Math. Soc. 131, No. 8, 2337--2346 (2003; Zbl 1017.13009)] showed that for \(D\) an integral domain with quotient field \(K\), \(D\subseteq K\) has FICP if and only if \(D\subseteq D^{\prime }\) and \(D^{\prime }\subseteq K\) have FICP (where \(D^{\prime }\) is the integral closure of \(D\)\ in \(K\)) if and only if \(D^{\prime }\) is a Prüfer domain with a finite number of prime ideals. The main result of the paper under review is the following generalization of Gilmer's result. Theorem: Let \( R\subseteq S\) be an extension of integral domains and let \(R^{\ast }\) be the integral closure of \(R\) in \(S\). Suppose that \(R^{\ast }\) is finite dimensional semiquasilocal. If \(R\subseteq R^{\ast }\) and \(R^{\ast }\subseteq S\) have FICP, then so does \(R\subseteq S\). An example using pullbacks is given where \(R\subset S\) has FICP, but \(R^{\ast }\) is not a Prüfer domain.
scientific article; zbMATH DE number 5602534
scientific article