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 S-primary if there exists an s∈S such that for all a,b∈R if ab∈P then sa∈P or sb∈P. We show that S-primary ideals enjoy analogs of many properties of primary ideals and we study the form of S-primary ideals of the amalgamation of A with B along an ideal J with respect to f (denoted by A ⋈f J), introduced and studied by D’Anna et al. S-primary ideals of the form I(+)M of the trivial ring extensions and S-primary ideals of the form I ⋈f,g(J,J′) and (K,L)f,g of the bi-amalgamations A ⋈f,g(J,J′) are characterized.