English

Abstract

Let R be a commutative ring with unity. The notion of maximal non λ-subrings is introduced and studied. A ring R is called a maximal non λ-subring of a ring T if R ⊂ T is not a λ-extension, and for any ring S such that R ⊂ S ⊆ T, S ⊆ T is a λ-extension. We show that a maximal non λ-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 λ-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non λ-subring. If R is a maximal non λ-subring of a field K, where R is integrally closed in K, then K is the quotient field of R and R is a Prufer domain. The equivalence of a maximal non λ-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 λ-subrings of a field.