Pointwise minimal extensions
The notion of minimal extension of rings was introduced by \textit{D. Ferrand} et al. [J. Algebra. 16, No. 3, 461--471 (1970; Zbl 0218.13011)]. \textit{P.-J. Cohen} et al. [J. Algebra Appl. 9, No. 1, 43--72 (2010; Zbl 1191.13009)] introduced pointwise minimal extensions, in the special context of domains. An extension \(R\subset S\) is said to be pointwise minimal extension if every simple sub-extension \(R\subset R[t]\) is minimal. The minimal pointwise extension \(R\subset S\) is called pointwise minimal pair if, for each \(T\) between \(R\) and \(S\), \(T\subset S\) is a pointwise minimal extension. The main goal of the paper under review is to characterized pointwise minimal extensions and pointwise minimal pair. The authors show first that pointwise minimal extension is either integral or integral closed. In Section 4, they focus on extensions over a field \(k\) and prove that there are four types of minimal extensions: decomposed, inert, ramified, and composite. In Section 5, the authors turn back to general integral extensions and fully characterize each type of pointwise minimal extensions (decomposed, inert, ramified, and composite). Numerous examples are contained in the last section of the paper.
scientific article; zbMATH DE number 6989342