Maximal subrings of classical integral domains

Abstract

It is shown that if R is an integral domain with │R│ = 2^c, 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 _eld, 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.