On the commutative ring extensions with at most two non Prüfer intermediate rings (Q2029733)

The rings under consideration are all commutative, with non-zero identity; and all the related objects are unital. Given a ring extension \(R\subset S\) and a ring-theoretic property \(\mathcal{P}\), denote by \([R,S]\) (\([R,S]_{\mathcal{P}}\), respectively) the set of rings contained between \(R\) and \(S\) (the set of rings between \(R\) and \(S\) with property \(\mathcal{P}\), respectively). Of interest to the authors is the set \([R,S] \subset [R,S]_{\mathcal{P}}\), denoted by \([R,S]_{\text{non-}\mathcal{P}}\) ; to abbreviate further, we denote it by ``not\(\mathcal{P}\)'', for a fixed extension \(R\subset S\) and property \(\mathcal{P}\). The authors find it interesting to characterize ring extensions \(R\subset S\) with \(|\text{not}\mathcal{P}|=n\) and in particular, when property \(\mathcal{P}\) is being a Prüfer domain (or a ring). The second author answered the question for \(n=0\) in \textit{Pairs of domains where most of the intermediate domains are Prüfer} [\textit{N. Jarboui}, J. Algebra Appl. 20, No. 6, Article ID 2150101, 13 p. (2021; Zbl 1465.13016)]. This paper gives the desired characterizations for integral domains, for \(n=1\) and \(n=2\). One of the results is that \(|\text{not}\mathcal{P}|=1\) if and only if \(R\) is a maximal non-Prüfer domain subring of \(S\), with other equivalences in terms of minimal overrings and minimal extensions. For integral domains, one consequence is that if \(|\text{not}\mathcal{P}|=n\) and \(n\geq 1\), then \(R\) is not a Prüfer domain. The case \(|\text{not}\mathcal{P}|=2\), for domains is characterized by exactly three possibilities in terms of minimal extensions and overrings (one case: \(R\subset S\) is a minimal extension of non-Prüfer domains); a corollary is made in case \(S\) is a field. Similar results are exhibited in case of Prüfer rings where \(R\) and \(S\) have some special properties. Then pullback construction is used to exhibit some further examples. Final section answers a question posed earlier by the second author and \textit{S. Aljubran} [``Maximal non-integrally closed subrings of an integral domain'', Ric. Mat. (to appear), \url{doi:10.1007/s11587-020-00500-0}].