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, 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.

Key Words:

• Almost prime ideal
• Prime ideal
• φ-Prime ideal
• Weakly prime ideal

2000 Mathematics Subject Classification:

• 13A15

Notes

Communicated by I. Swanson.

Additional information

Notes on contributors

Malik Bataineh

*New affiliation: Mathematics Department, The Hashemite University, Zarqa, Jordan.