A note on -domains and -domains

Abstract

Let R be an integral domain. Then R is said to be a λ-domain if the set of all overrings of R is linearly ordered by inclusion. If R1+R2 is an overring of R for each pair of overrings R1,R2 of R, then R is said to be a Δ-domain. We show that if R⊂T is an extension of integral domains such that each proper subring of T containing R is a λ-domain (resp., Δ-domain), then T is a λ-domain (resp., Δ-domain under some conditions). Moreover, the pair (R,T) is a residually algebraic pair. Two new ring theoretic properties, namely λ-property of domains and Δ-property of domains are introduced and studied.