On weakly S-primary ideals of commutative rings

Abstract

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.