<i>S</i>-primary ideals of a commutative ring

Massaoud, Essebti

doi:10.1080/00927872.2021.1977939

Cited by 12 publications

(8 citation statements)

References 5 publications

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“…Let S ⊆ R be a multiplicative set and P be an ideal of R such that P ∩ S = ∅. For an ideal P of R, the radical of P denoted by rad(P ) is the ideal {x ∈ R : x n ∈ P for some positive integer n } of R. In [8], P is said to be an S-primary ideal of R if there exists s ∈ S such that for all x, y ∈ R, if xy ∈ P , then sx ∈ P or sy ∈ rad(P ). P is said to be a weakly S-primary ideal of R if there exists s ∈ S such that for all x, y ∈ R, if 0 = xy ∈ P , then sx ∈ P or sy ∈ rad(P ) (see [6]).…”

Section: Introductionmentioning

confidence: 99%

Generalizations of Graded $S$-Primary Ideals

Al-Shorman¹,

Bataineh²,

Abu-Dawwas³

2022

Preprint

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The goal of this article is to present the graded weakly S-primary ideals and g-weakly S-primary ideals which are extensions of graded weakly primary ideals. Let R be a commutative graded ring, S ⊆ h(R) and P be a graded ideal of R. We state P is a graded weakly S-primary ideal of R if there exists s ∈ S such that for all x, y ∈ h(R), if 0 = xy ∈ P , then sx ∈ P or sy ∈ Grad(P ) (the graded radical of P ). Several properties and characteristics of graded weakly S-primary ideals as well as graded g-weakly S-primary ideals are investigated.Grad(P ) = x = g∈G x g ∈ R : for all g ∈ G, there exists n g ∈ N such that x g ng ∈ P .Note that Grad(P ) is always a graded ideal of R (see [10]).Let P be a graded ideal of R and P = R, then P is said to be graded primary ideal of R, if x, y ∈ h(R) with xy ∈ P then x ∈ P or y ∈ Grad(P ) (see [12]). The concept of a graded primary ideal can be generalized in a many ways, for example, Bataineh in [5] and Atani in [3]. A proper graded ideal P of R is claimed to be a graded weakly primary ideal if whenever 0 = xy ∈ P for some x, y ∈ h(R), then either x ∈ P or y ∈ Grad(P ).

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Section: Introductionmentioning

confidence: 99%

Generalizations of Graded $S$-Primary Ideals

Al-Shorman¹,

Bataineh²,

Abu-Dawwas³

2022

Preprint

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“…A proper ideal P of R disjoint from S is called an S-prime of R if there exists an s ∈ S such that for all x, y ∈ R if xy ∈ P , then sx ∈ P or sy ∈ P . In [24] , Massaoud defined and investigated the concept of S-primary ideals of a commutative ring in a way that generalizes essentially all the results concerning primary ideals. A proper ideal Q of R disjoint from S is called an S-primary of R if there exists an s ∈ S such that for all x, y ∈ R if xy ∈ P , then sx ∈ P or sy ∈ √ Q.…”

Section: Introductionmentioning

confidence: 99%

Extensions of $n$-ary prime hyperideals via an $n$-ary multiplicative subset in a Krasner $(m,n)$-hyperring

Anbarloei¹

2022

Preprint

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Let R be a Krasner (m, n)-hyperring and S be an n-ary multiplicative subset of R. The purpose of this paper is to introduce the notion of n-ary S-prime hyperideals as a new expansion of n-ary prime hyperideals. A hyperideal I of R disjoint with S is said to be an n-ary S-prime hyperideal if there exists s ∈ S such that whenever g(x n 1 ) ∈ I for all x n 1 ∈ R, then g(s, x i , 1 (n−2) ) ∈ I for some 1 ≤ i ≤ n. Several properties and characterizations concerning n-ary S-prime hyperideals are presented. The stability of this new concept with respect to various hyperring-theoretic constructions are studied. Furthermore, we extend this concept to n-ary S-primary hyperideals. We obtained some specific results explaining the structure.2010 Mathematics Subject Classification. 16Y99. Key words and phrases.n-ary J-hyperideal, n-ary δ-J-hyperideal, (k, n)-absorbing δ-Jhyperideal.

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“…In [3], S. E. Atani and F. Farzalipour have defined a proper ideal of R to be weakly primary if 0 ̸ = ab ∈ P implies a ∈ P or b ∈ √ P . The first author in [12] introduced and investigated the concept of S-primary ideals which constitute a generalization of primary ideals. More precisely, let R be a commutative ring, S a multiplicative subset of R and I an ideal of R disjoint from S. Then, I is called an S-primary ideal of R if there exists an s ∈ S such that for all a, b ∈ R if ab ∈ I, then sa ∈ I or sb ∈ √ I.…”

Section: Introductionmentioning

confidence: 99%

Some results about weakly S-primary ideals of a commutative ring

Massaoud¹,

Gouaid²

2022

GJOM

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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 weakly S-primary if there exists an s ∈ S such that for all a, b ∈ R if 0≠ ab ∈ P then sa ∈ P or sb ∈ √P. We show that weakly S-primary ideals enjoy analogs of many properties of weakly primary ideals and we study the form of weakly S-primary ideals of the amalgamation of A with B along an ideal J with respect to f (denoted by A ⋈fJ). Weakly S-primary ideals of the trivial ring extension are also characterized.

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S-primary ideals of a commutative ring

Cited by 12 publications

References 5 publications

Generalizations of Graded $S$-Primary Ideals

Generalizations of Graded $S$-Primary Ideals

Extensions of $n$-ary prime hyperideals via an $n$-ary multiplicative subset in a Krasner $(m,n)$-hyperring

Some results about weakly S-primary ideals of a commutative ring

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