scholarly journals On Weakly S-Primary Ideals of Commutative Rings

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.

scholarly journals On 1-absorbing primary ideals of commutative rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050111 ◽  
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]

On weakly 2-absorbing δ-primary ideals of commutative rings

2020 ◽  
Vol 27 (4) ◽  
pp. 503-516
Author(s):  
Ayman Badawi ◽  
Brahim Fahid
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Primary Ideal ◽  
New Class ◽  
Primary Ideals

AbstractLet R be a commutative ring with {1\neq 0}. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever {a,b,c\in R} and {0\not=abc\in I}, then {ab\in I} or {ac\in\sqrt{I}} or {bc\in\sqrt{I}}. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let {I(R)} be the set of all ideals of R and let {\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever {L,I,J} are ideals of R with {J\subseteq I}, then {L\subseteq\delta(L)} and {\delta(J)\subseteq\delta(I)}. Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., {I\not=R}) is called a weakly 2-absorbing δ-primary ideal if {0\not=abc\in I} implies {ab\in I} or {ac\in\delta(I)} or {bc\in\delta(I)}. For example, let {\delta:I(R)\rightarrow I(R)} such that {\delta(I)=\sqrt{I}}. Then δ is an expansion function of ideals of R, and hence a proper ideal I of R is a weakly 2-absorbing primary ideal of R if and only if I is a weakly 2-absorbing δ-primary ideal of R. A number of results concerning weakly 2-absorbing δ-primary ideals and examples of weakly 2-absorbing δ-primary ideals are given.

ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

2021 ◽  
Vol 10 (11) ◽  
pp. 3479-3489
Author(s):  
K. Al-Zoubi ◽  
M. Al-Azaizeh
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Closed Subset ◽  
Commutative Rings ◽  
Graded Modules ◽  
Prime Submodules ◽  
Prime Submodule

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

scholarly journals On graded primary-like submodules of graded modules over graded commutative rings

2021 ◽  
Author(s):  
Khaldoun Falah Al-Zoubi ◽  
Mohammed Al-Dolat
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Graded Modules ◽  
Primary Ideals

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.

Strongly ϕ-n-irreducible ideals

2021 ◽  
pp. 2350002
Author(s):  
Ahmad Yousefian Darani ◽  
Najib Mahdou ◽  
Sanae Moussaoui
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Ring Extension ◽  
The Stability

Let [Formula: see text] be a commutative ring, [Formula: see text] be an ideal of [Formula: see text], [Formula: see text] be a non-null positive integer and [Formula: see text] be a function where [Formula: see text] is the set of ideals of [Formula: see text]. In this paper, we define a new generalization of strongly [Formula: see text]-irreducible ideals called strongly [Formula: see text]-[Formula: see text]-irreducible ideal, that is, whenever [Formula: see text] and [Formula: see text] for [Formula: see text] ideals of [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose intersection is in [Formula: see text]. We study 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.

scholarly journals On 2-Absorbing Primary Submodules of Modules over Commutative Rings

2016 ◽  
Vol 24 (1) ◽  
pp. 335-351 ◽  
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.

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

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.

Primary Ideals and Prime Power Ideals

1966 ◽  
Vol 18 ◽  
pp. 1183-1195 ◽  
Author(s):  
H. S. Butts ◽  
Robert W. Gilmer
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Prime Power ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Primary Ideal ◽  
Primary Ideals ◽  
The Ideal

This paper is concerned with the ideal theory of a commutative ringR.We sayRhas Property (α) if each primary ideal inRis a power of its (prime) radical;Ris said to have Property (δ) provided every ideal inRis an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that ifDis a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, thenDis a Dedekind domain. It follows from this that ifDhas Property (α) and is Noetherian (in which caseDhas Property (δ)), thenDis Dedekind.

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

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.

scholarly journals On 1-absorbing δ-primary ideals

2021 ◽  
Vol 29 (3) ◽  
pp. 135-150
Author(s):  
Abdelhaq El Khalfi ◽  
Najib Mahdou ◽  
Ünsal Tekir ◽  
Suat Koç
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Basic Properties ◽  
New Class ◽  
Ring Extensions ◽  
Primary Ideals ◽  
Nonzero Identity

Abstract Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

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