On Generalizations of Prime Ideals

Abstract

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {∅} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n − 1, n) − φ-prime if, whenever a ₁, a ₂, ⋅, a _n  ∈ R and a ₁ a ₂⋅a _n  ∈ I∖φ(I), the product of (n − 1) of the a _i 's is in I. In this article, we study (n − 1, n) − φ-prime ideals (n ≥ 2). A number of results concerning (n − 1, n) − φ-prime ideals and examples of (n − 1, n) − φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n − 1, n) − φ-prime, are characterized.

Key Words:

• Local ring
• (n − 1
• n) − n-Almost prime ideal
• (n − 1
• n) − φ-Prime ideal
• Von Neumann regular ring
• (n − 1
• n)-Weakly prime ideal

2000 Mathematics Subject Classification:

• 13A15

ACKNOWLEDGMENTS

The authors would like to thank the referee for his/her useful suggestions that improved the presentation of this article.

This research has been supported by the Linear Algebra and Optimization Center of Excellence of Shahid Bahonar University of Kerman.

Notes

Communicated by I. Swanson.