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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 ∈ S such that whenever 0 ≠ a b ∈ I for a , b ∈ R , then s a ∈ I or s b ∈ I. The relationships among S -prime, S -primary, weakly S -primary and S - n -ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S r -primary ideals are characterized where S r = { 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 S -decomposable ideals and S -Laskerian rings are introduced. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal I of R is called a 1 -absorbing primary ideal of R if whenever nonunit elements a , b , c ∈ R and a b c ∈ I , then a b ∈ I or c ∈ I. Some properties of 1-absorbing primary ideals are investigated. For example, we show that if R admits a 1-absorbing primary ideal that is not a primary ideal, then R is a quasilocal ring. We give an example of a 1-absorbing primary ideal of R that is not a primary ideal of R. We show that if R is a Noetherian domain, then R is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of R is of the form P n for some nonzero prime ideal P of R and a positive integer n ≥ 1. We show that a proper ideal I of R is a 1-absorbing primary ideal of R if and only if whenever I 1 I 2 I 3 ⊆ I for some proper ideals I 1 , I 2 , I 3 of R , then I 1 I 2 ⊆ I or I 3 ⊆ I. [ABSTRACT FROM AUTHOR]