Weakly α-prime ideals

In this paper, we introduce and study the notion of weakly [Formula: see text]-prime ideals. Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text]. We say that [Formula: see text] is a weakly [Formula: see text]-prime ideal if whenever [Formula: see text], then [Formula: see text] or [Formula: see text], equivalently [Formula: see text] or [Formula: see text]. We show that if [Formula: see text], then the notion of [Formula: see text]-prime and weakly [Formula: see text]-prime ideals coincide. We give several basic properties of the notion of weakly [Formula: see text]-prime ideals and we study its transfer to trivial ring extensions and amalgamations of rings. This allows us to construct nontrivial examples of weakly [Formula: see text]-prime ideals that are neither [Formula: see text]-prime nor weakly prime ideals.