When only finitely many intermediate rings result from juxtaposing two minimal ring extensions
A ring extension \(A\subsetneq B\) of commutative rings is a \textit{minimal extension} of \(A\) if there is no ring \(C\) such \(A\subsetneq C \subsetneq B\). Clearly, a minimal ring extension is either integrally closed or integral. For rings \(R\subsetneq S\subsetneq T\) such that \(R\subsetneq S\) and \(S\subsetneq T\) are minimal ring extensions, the authors present necessary and sufficient conditions for the extension \(R\subsetneq T\) to have only finitely many intermediate rings (condition FIP). For example, by a previous result of the present authors, if the extension \(S\subsetneq T\) is integraly closed, then the extension \(R\subsetneq T\) satisfies FIP. To characterize the FIP property of the extension \(R\subsetneq T\) the authors take into account the possible three properties of an integral minimal extension (inert, ramified and decomposed. The definitions are recalled in the paper.) Examples are provided for a full analysis of all the possibilities. Moreover, using this analysis, the authors study the case when the extension \(R\subsetneq T\) contains just one proper intermediate ring. This study is completed by the first author in the paper [Palest. J. Math. 6, No. 1, 31--44 (2017; Zbl 1354.13014)].
scientific article; zbMATH DE number 6633367