λ-rings, φ-λ-rings, and φ-Δ-rings

Abstract

Let R be a commutative ring with unity. The notion of λ-rings, φ-λ-rings, and φ-Δ-rings is introduced which generalize the concept of λ-domains and Δ-domains. A ring R is said to be a λ-ring if the set of all overrings of R is linearly ordered under inclusion. A ring R in H is said to be a φ-λ-ring if φ(R) is a λ-ring, and a φ-Δ-ring if φ(R) is a Δ-ring, where H is the set of all rings such that Nil(R) is a divided prime
ideal of R and φ: T(R) to R_Nil(R) is a ring homomorphism defined as φ(x) = x for all x in T(R). The equivalence of φ-λ-rings, φ-Δ-rings with the latest trending rings in the literature, namely, φ-chained rings and φ-Prufer rings is established under some conditions. Using the idealization theory of Nagata, examples are also given to strengthen the concept.