On weakly S-primary ideals of commutative rings

Let Formula: see text be a commutative ring with identity and Formula: see text be a multiplicatively closed subset of Formula: see text. The purpose of this paper is to introduce the concept of weakly Formula: see text-primary ideals as a new generalization of weakly primary ideals. An ideal Formula: see text of Formula: see text disjoint with Formula: see text is called a weakly Formula: see text-primary ideal if there exists Formula: see text such that whenever Formula: see text for Formula: see text, then Formula: see text or Formula: see text. The relationships among Formula: see text-prime, Formula: see text-primary, weakly Formula: see text-primary and Formula: see text-Formula: see text-ideals are investigated. For an element Formula: see text in any general ZPI-ring, the (weakly) Formula: see text-primary ideals are characterized where Formula: see text. Several properties, characterizations and examples concerning weakly Formula: see text-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly Formula: see text-decomposable ideals and Formula: see text-weakly Laskerian rings which are generalizations of Formula: see text-decomposable ideals and Formula: see text-Laskerian rings are introduced.