2019•FilomatOpen access

λ-rings, Φ-λ-rings, and Φ-Δ-rings

Rahul Kumar, Atul Gaur

Abstract

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

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Key concepts: Mathematics, Principal ideal ring, Noncommutative ring, Reduced ring, Ring (chemistry), Primitive ring, Von Neumann regular ring, Commutative ring