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 epsilon R with ab epsilon I - {0} (resp., ab epsilon I - I-2) implies a epsilon I or b epsilon I. Let 0 : F (R) -> F (R) U {phi} be a function where F (R) is the set of ideals of R. We call a proper ideal I of R a phi-prime ideal if a, b epsilon R with ab epsilon I - phi(I) implies a epsilon I or b epsilon 1. So taking phi(empty set)(J) = empty set (resp., phi(0) (J) = 0, phi(2) (J) = J(2)), a phi(empty set)-prime ideal (resp., phi(0)-prime ideal, phi(2)-prime ideal) is a prime ideal (resp., weakly prime ideal almost prime ideal). We show that phi-prime ideals enjoy analogs of many of the properties of prime ideals.