Some results about weakly S-primary ideals of a commutative ring

Let R be a commutative ring with identity and S ⊊ R a multiplicative subset. We define a proper ideal P of R disjoint from S to be weakly S-primary if there exists an s ∈ S such that for all a, b ∈ R if 0≠ ab ∈ P then sa ∈ P or sb ∈ √P. We show that weakly S-primary ideals enjoy analogs of many properties of weakly primary ideals and we study the form of weakly S-primary ideals of the amalgamation of A with B along an ideal J with respect to f (denoted by A ⋈fJ). Weakly S-primary ideals of the trivial ring extension are also characterized.