Ricerche mat. https://doi.org/10.1007/s11587-018-0388-6 Maximal non-prime ideally equal subrings of a commutative ring Naseam Al-Kuleab 1 · Noômen Jarboui 1 · Almallah Omar 2 Received: 23 November 2017 / Revised: 8 March 2018 © Università degli Studi di Napoli "Federico II" 2018 Abstract A commutative ring R is said to be maximal non-prime ideally equal subring of S, if Spec( R)  = Spec( S), whereas Spec(T ) = Spec( S) for any subring T of S properly containing R. The aim of this paper is to give a complete characterization of this class of rings. Keywords Integral domain · Prime ideal · Algebraic extension · Residually algebraic pair · Prüfer domain · Valuation domain · Pullback Mathematics Subject Classification 13A15 · 13B25 · 13C15 1 Introduction All rings considered in this paper are commutative and unital; all inclusions of rings are unital ring extensions. If R is a commutative ring, we use Spec( R) to denote the set of prime ideals of R, Max( R) for its set of maximal ideals. Let P be a property of Communicated by M.Fontana. B Noômen Jarboui njarboui@kfu.edu.sa Naseam Al-Kuleab naalkleab@kfu.edu.sa Almallah Omar oamallah@bau.edu.jo 1 Department of Mathematics, Faculty of Sciences, King Faisal University, P.O. Box 380, Al-hassa 31982, Saudi Arabia 2 Al-Balqa University, Amman, Jordan 123