Generalizations of prime ideals

Abstract

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, bR with abI-{0} (resp., abI-I2) implies aI or bI. Let :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 -prime ideal if a, bR with abI-(I) implies aI or bI. So taking (J)= (resp., 0(J)=0, 2(J)=J2), a -prime ideal (resp., 0-prime ideal, 2-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that -prime ideals enjoy analogs of many of the properties of prime ideals.