On λ -extensions of commutative rings

Abstract

Let R,T be commutative rings with identity such that R⊆T. We recall that R⊆T is called a λ-extension of rings if the set of all subrings of T containing R (the “intermediate rings”) is linearly ordered under inclusion. In this paper, a characterization of integrally closed λ-extension of rings is given. For example, we show that if R is a local ring, then R⊆T is an integrally closed λ-extension of rings if and only if there exists q∈Spec(R) such that T=Rq,q=Tq and R/q is a valuation domain. Let R be a subring of T such that R is invariant under action by G, where G is a subgroup of the automorphism group of T. If R⊆T is a λ-extension of rings, then RG⊆TG is a λ-extension of rings under some conditions.