Open Accessλ-rings, Φ-λ-rings, and Φ-Δ-rings
Author(s) -
Rahul Kumar,
Atul Gaur
Publication year - 2019
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1916125k
Subject(s) - mathematics , principal ideal ring , noncommutative ring , reduced ring , ring (chemistry) , primitive ring , von neumann regular ring , commutative ring , category of rings , radical of a ring , pure mathematics , semiprime ring , morita equivalence , homomorphism , boolean ring , combinatorics , prime (order theory) , commutative property , chemistry , organic chemistry
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.