Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I − {0} (resp., ab ∈ I − I 2) implies a ∈ I or b ∈ I. Let φ:ℐ(R) → ℐ(R) ∪ {∅} be a function where ℐ(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I − φ(I) implies a ∈ I or b ∈ I. So taking φ∅(J) = ∅ (resp., φ0(J) = 0, φ2(J) = J 2), a φ∅-prime ideal (resp., φ0-prime ideal, φ2-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.