An answer to a question about maximal non-integrally closed subrings

The main purpose of this paper is to answer a question which was left open in N. Jarboui and S. Aljubran [Maximal non-integrally closed subrings of an integral domain, Ric. Mat. (2020), doi.org/10.1007/s11587-020-00500-0] asking for a characterization of maximal non-integrally closed subrings of arbitrary rings.