Maximal non λ-subrings

Abstract

Let R be a commutative ring with unity. The notion of maximal non -subrings is introduced and studied. A ring R is called a maximal non -subring of a ring T if R T is not a -extension, and for any ring S such that R S T , S T is a -extension. We show that a maximal non -subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non -domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non -subring. If R is a maximal non -subring of a field K, where R is integrally closed in K, then K is the quotient field of R and R is a Prüfer domain. The equivalence of a maximal non -domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non -subrings of a field.