Weakly α-prime ideals

In this paper, we introduce and study the notion of weakly α-prime ideals. Let R be a commutative ring, I an ideal of R and α ∈ End(R). We say that I is a weakly α-prime ideal if whenever 0 ≠ ab ∈ I, then a ∈ I or α(b) ∈ I, equivalently b ∈ I or α(a) ∈ I. We show that if I² ≠ 0, then the notion of α-prime and weakly α-prime ideals coincide. We give several basic properties of the notion of weakly α-prime ideals and we study its transfer to trivial ring extensions and amalgamations of rings. This allows us to construct nontrivial examples of weakly α-prime ideals that are neither α-prime nor weakly prime ideals.