On Generalizations of Prime Ideals (II)

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.