On weakly S-primary ideals of commutative rings

Abstract

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s is an element of S such that wheneverv 0 not equal ab is an element of I for a, b is an element of R, then sa is an element of root I or sb is an element of I. The relationships among S-prime, S-primary, weakly S-primary and S-nideals are investigated. For an element r in any general ZPI-ring, the (weakly) Sr-primary ideals are characterized where Sr = {1, r, r(2),...}. Several properties, characterizations and examples concerning weakly S-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 S-decomposable ideals and S-weakly Laskerian rings which are generalizations of Sdecomposable ideals and S-Laskerian rings are introduced.