Maximal subrings of classical integral domains

It is shown that if R is an integral domain with |R| = 2ʿ, then either R has a maximal subring or R has a prime ideal P which is not a maximal ideal of R, Char(R/P) = Char(R) and |R/P| < |R|. We prove that if R = A ⊕ I is a ring, where A is a subring of R and I is an ideal of R and I/I2 is a finitely generated nonzero R/I-module, then R has a maximal subring. We show that if R is an infinite Noetherian ring which is not integral over its prime subring and R has no maximal subring, then R has a prime ideal P (which is not a maximal ideal of R) such that R/P is not integral over its prime subring but for each ideal I of R with P ⊊ I, R/I is integral over its prime subring, in particular dim(R/P ) = 1 or 2 and |R/P| = |R| is countable. The existence of maximal subrings in Prüfer and Bézout domains is investigated. Moreover, we prove that if R is a maximal subring of an integral domain T , where T is not a field, and R is a Prüfer domain, then R is integrally closed in T and consequently T is an overring of R and T is a Prüfer domain. Conversely, if R is integrally closed in T and T is a Prüfer domain, then R is a Prüfer domain. Finally, we study the existence of maximal subrings in a ring T by the use of survival pairs and valuation pairs too.