The study of either properties or the number of intermediate rings lying between a ring extension \(R\subseteq S\) has a historical background, as written in the introduction of the paper under review. For instance it is well-known that a (commutative unital) domain \(R\) is a Prüfer domain if and only if each intermediate ring between \(R\) and \(\mathrm{Frac}\;R\) is integrally closed in \(\mathrm{Frac}\;R\), that is to say \((R,\mathrm{Frac}\;R)\) is a normal pair if and only if \(R\) is Prüfer domain. On the other hand, for any normal pair \((R,S)\), it is proved that there exists only finite number of intermediate rings if and only if any chain of distinct intermediate rings is of finite length.
Let \(R^*\) denotes the integral closure of \(R\) in an extension \(S\) of \(R\). Suppose that \((R^*,S)\) is a normal pair and, furthermore, any chain of indeterminate rings lying between \(R\) and \(R^*\) has finite length. For such an extension \((R,S)\), as the main result of the second section of the paper, it is proved that there exists finite number of maximal ideals \(\mathfrak{m}_i\) of \(R\) such that any \(T\in [R,S]\) satisfying \(T\cap R^*=R\) can be described as \(\big(\cap\;R_{\mathfrak{m}_i}\big)\cap T'\) for some \(T'\in [R^*,S]\). As an immediate corollary to this Theorem, it can be deduced that if moreover \(R\subset R^*\) is a minimal extension and \(R\) is quasi-local then \(R^*\) is the least member of \([R,S]\backslash \{R\}\).
The third section of the paper is devoted to providing some necessary and sufficient conditions for an extension \(T\subset T'\), between \(R\) and \(S\), to be a minimal extension. The motivation of this study, as stated in the paper, is the fact that in any maximal chain of distinct intermediate rings, any two consecutive extension \(R_i\subseteq R_{i+1}\) has to be minimal.
The forth section is devoted to comparing the length of a maximal chain \(\mathcal{C}\) of intermediate rings in \([R,S]\) with the corresponding chain \(\mathcal{C}^*\), in \([R,R^*]\), obtained by intersecting with \(R^*\) and then eliminating the repetitions. Namely, assuming that the extension \(R\subseteq S\) admits only finite length chain of indeterminate rings, the authors show that the chain \(\mathcal{C}^*\) is also maximal and they establish the identity \(l(\mathcal{C})=l(\mathcal{C}^*)+|\mathrm{Supp}(S/R^*)|\).