On Generalizations of Prime Ideals (II)

Abstract

Let R be a commutative ring with identity. We say that a proper ideal P of R is (n − 1, n)-weakly prime (n ≥ 2) if 0 ≠ a ₁…a _n  ∈ P implies a ₁…a _i−1 a _i+1…a _n  ∈ P for some i ∈ {1,…, n}, where a ₁,…, a _n  ∈ R. In this article, we study (n − 1, n)-weakly prime ideals. A number of results concerning (n − 1, n)-weakly prime ideals and examples of (n − 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n − 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n − 1, n)-weakly prime ideals and (n − 1, n)-prime ideals coincide.

Key Words:

• Quasi-local ring
• (n − 1
• n)-Weakly prime ideal
• (n − 1
• n)-Prime ideal

2010 Mathematics Subject Classification:

• 13A15

ACKNOWLEDGMENT

I would like to thank the referee for a careful reading of my article and insightful comments which saved me from several errors and improved the presentation of this article.

Notes

Communicated by I. Swanson.