Abstract
In this paper we classify commutative ring extensions with exactly two non Prufer domain intermediate rings. An initial step involves the description of the commutative ring extensions with only one non Prufer domain intermediate ring. Some generalizations to the context of rings with zero divisors are proved. We also answer a question which was left open in Jarboui and Aljubran (Ric Mat, https://doi.org/10.1007/s11587-020-00500-0, 2020). More precisely, let S=K[y1]]...[yt]] be a K-algebra (not necessarily finitely generated over the field K) having Krull dimension n >= 1. Let I be a nonzero proper ideal of S (not necessarily maximal in S) and D be a proper subring of K. We provide necessary and sufficient conditions in order that R=D+I is a maximal non-integrally closed subring of S.