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  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) → RNil(R) is a ring homomorphism defined as Φ(x) = x for all x  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-Prüfer rings is established under some conditions. Using the idealization theory of Nagata, examples are also given to strengthen the concept.