A note on ϕ -λ -rings and ϕ -Δ -rings

Abstract

Let H denotes the set of all commutative rings R in which the set of all nilpotent elements, denoted by Nil(R), is a prime ideal of R and is comparable to every ideal of R. Let R∈H be a ring and T(R) be its total quotient ring. Then there is a ring homomorphism ϕ:T(R)→RNil(R) defined as ϕ(r/s)=r/s for all r∈R and for all non-zero-divisors s∈R. A ring R∈H is said to be a ϕ-λ-ring if the set of all rings between ϕ(R) and T(ϕ(R)) is linearly ordered by inclusion. If R1+R2 is a ring between ϕ(R) and T(ϕ(R)) for each pair of rings R1,R2 between ϕ(R) and T(ϕ(R)), then R is said to be a ϕ-Δ-ring. Let R∈H be a ϕ-λ-ring and T∈H be a ring properly containing R such that Nil(T)=Nil(R). We show that if all but finitely many intermediate rings between R and T are ϕ-λ-rings (resp., ϕ-Δ-rings), then all the intermediate rings are ϕ-λ-rings (resp., ϕ-Δ-rings under some conditions). Moreover, the pair (R, T) is a residually algebraic pair. Two new ring theoretic properties, namely, ϕ-λ-property of rings and ϕ-Δ-property of rings are introduced and studied.