J-prime ideals of commutative rings

Let R be a commutative ring with identity, and J(R) denote the Jacobson radical of R. This paper introduces J-prime ideals, generalizing prime ideals, n-ideals, and J-ideals. A proper ideal I of R is a J-prime ideal if for every a, b ∈ R, ab ∈ I implies a ∈ I + J(R) or b ∈ I. We characterize rings in which every proper ideal is J-prime, showing that a ring has the property that every proper ideal is J-prime if and only if it is a quasilocal ring. Also, we show that (0) is a J-prime ideal if and only if the ring is présimplifiable. Furthermore, we examine J-prime ideal characteristics in various ring constructions, such as homomorphic image of rings, quotient rings, cartesian product rings, polynomial rings, power series rings, trivial ring extension and amalgamation rings.