GENERALIZED OF PRIMARY AVOIDANCE THEOREM  FOR COMMUTATIVE RINGS

Let G be a group with identity e. Let R be a G-graded commutative ring andM a graded R-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to and extra properties of these graded submodules.

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On 1-absorbing primary ideals of commutative rings

Journal of Algebra and Its Applications ◽

10.1142/s021949882050111x ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050111 ◽

Cited By ~ 2

Author(s):

Ayman Badawi ◽

Ece Yetkin Celikel

Keyword(s):

Positive Integer ◽

Prime Ideal ◽

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Primary Ideal ◽

Ideal Formula ◽

Noetherian Domain ◽

Primary Ideals ◽

Nonzero Identity

Let [Formula: see text] 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 [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

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On 2-Absorbing Primary Submodules of Modules over Commutative Rings

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.1515/auom-2016-0020 ◽

2016 ◽

Vol 24 (1) ◽

pp. 335-351 ◽

Cited By ~ 1

Author(s):

Hojjat Mostafanasab ◽

Ece Yetkin ◽

Ünsal Tekir ◽

Ahmad Yousefian Darani

Keyword(s):

Commutative Rings ◽

Primary Ideal ◽

Finitely Generated ◽

Primary Submodules ◽

Primary Submodule ◽

Prime Submodule ◽

Primary Ideals

Abstract All rings are commutative with 1 ≠ 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a; b ∈ R and m ∈ M and abm ∈ N, then am ∈ M-rad(N) or bm ∈ M-rad(N) or ab ∈(N :R M). It is shown that a proper submodule N of M is a 2-absorbing primary submodule if and only if whenever I1I2K ⊆ N for some ideals I1; I2 of R and some submodule K of M, then I1I2 ⊆ (N :R M) or I1K ⊆ M-rad(N) or I2K ⊆ M-rad(N). We prove that for a submodule N of an R-module M if M-rad(N) is a prime submodule of M, then N is a 2-absorbing primary submodule of M. If N is a 2-absorbing primary submodule of a finitely generated multiplication R-module M, then (N :R M) is a 2-absorbing primary ideal of R and M-rad(N) is a 2-absorbing submodule of M.

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Properties of ϕ-δ-primary and 2-absorbing δ-primary ideals of commutative rings

Asian-European Journal of Mathematics ◽

10.1142/s1793557120500266 ◽

2018 ◽

Vol 13 (01) ◽

pp. 2050026

Author(s):

Ameer Jaber

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Expansion Formula ◽

Primary Ideals ◽

The Ideal

Let [Formula: see text] be a commutative ring with unity [Formula: see text] and let [Formula: see text] be an ideal expansion. In the first part of this paper, we extend the concept of [Formula: see text]-primary ideals to [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction and [Formula: see text] is an ideal expansion. We introduce some of the ideal expansion [Formula: see text] and define [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction. Also, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized [Formula: see text]-[Formula: see text]-primary ideals with respect to such an ideal expansion [Formula: see text]. In the second part of this paper we investigate 2-absorbing [Formula: see text]-primary ideals which unify 2-absorbing ideals and 2-absorbing primary ideals, where [Formula: see text] is an ideal expansion. A number of results in the two parts are given.

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On Weakly S-Primary Ideals of Commutative Rings

10.20944/preprints202109.0486.v1 ◽

2021 ◽

Author(s):

Ece Yetkin Celikeli ◽

Hani Khashan

Keyword(s):

Commutative Ring ◽

Closed Subset ◽

Commutative Rings ◽

Primary Ideal ◽

Ring Extension ◽

The Stability ◽

Primary Ideals

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≠ab∈I for a,b∈R, then sa∈√I or sb∈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 charctarized where S={1,r,r²,⋯}. 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.

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CLASSICAL PRIMARY SUBMODULES AND DECOMPOSITION THEORY OF MODULES

Journal of Algebra and Its Applications ◽

10.1142/s0219498809003369 ◽

2009 ◽

Vol 08 (03) ◽

pp. 351-362 ◽

Cited By ~ 6

Author(s):

M. BAZIAR ◽

M. BEHBOODI

Keyword(s):

Existence And Uniqueness ◽

Commutative Rings ◽

Finitely Generated ◽

Decomposition Theory ◽

One Dimensional ◽

Primary Submodules ◽

Primary Ideals ◽

Finitely Generated Modules

We introduce the notion of classical primary submodules that generalizes the concept of primary ideals of commutative rings to modules. Existence and uniqueness of classical primary decompositions in finitely generated modules over one-dimensional Noetherian domains are proved.

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The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings

Tatra Mountains Mathematical Publications ◽

10.2478/tmmp-2019-0015 ◽

2019 ◽

Vol 74 (1) ◽

pp. 7-16

Author(s):

Khaldoun Al-Zoubi ◽

Malik Jaradat

Keyword(s):

Commutative Rings ◽

Primary Ideal ◽

Zariski Topology ◽

Graded Ring ◽

Primary Spectrum ◽

Primary Ideals

Abstract Let G be a group with identity e and let R be a G-graded ring. A proper graded ideal P of R is called a graded primary ideal if whenever rgsh∈P, we have rg∈ P or sh∈ Gr(P), where rg,sg∈ h(R). The graded primary spectrum p.Specg(R) is defined to be the set of all graded primary ideals of R.In this paper, we define a topology on p.Specg(R), called Zariski topology, which is analogous to that for Specg(R), and investigate several properties of the topology.

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