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2-absorbing δ-semiprimary Ideals of Commutative Rings
Celikel, Ece Yetkin Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.4
Let R be a commutative ring with nonzero identity, 𝓘(𝓡) the set of all ideals of R and δ : 𝓘(𝓡) → 𝓘(𝓡) an expansion of ideals of R. In this paper, we introduce the concept of 2-absorbing δ-semiprimary ideals in commutative rings which is an extension of 2-absorbing ideals. A proper ideal I of R is called 2-absorbing δ-semiprimary ideal if whenever a, b, c ∈ R and abc ∈ I, then either ab ∈ δ(I) or bc ∈ δ(I) or ac ∈ δ(I). Many properties and characterizations of 2-absorbing δ-semiprimary ideals are obtained. Furthermore, 2-absorbing δ<sub>1</sub>-semiprimary avoidance theorem is proved.
On 2-absorbing primary ideals in commutative rings
Ayman Badawi,Unsal Tekir,Ece Yetkin 대한수학회 2014 대한수학회보 Vol.51 No.4
Let R be a commutative ring with 1 6= 0. In this paper, we introduce the concept of 2-absorbing primary ideal which is a general- ization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ √I or bc ∈ √I. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.
ON 2-ABSORBING PRIMARY IDEALS IN COMMUTATIVE RINGS
Badawi, Ayman,Tekir, Unsal,Yetkin, Ece Korean Mathematical Society 2014 대한수학회보 Vol.51 No.4
Let R be a commutative ring with $1{\neq}0$. In this paper, we introduce the concept of 2-absorbing primary ideal which is a generalization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever $a,b,c{\in}R$ and $abc{\in}I$, then $ab{\in}I$ or $ac{\in}\sqrt{I}$ or $bc{\in}\sqrt{I}$. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.
ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS
Badawi, Ayman,Tekir, Unsal,Yetkin, Ece Korean Mathematical Society 2015 대한수학회지 Vol.52 No.1
Let R be a commutative ring with $1{\neq}0$. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, $c{\in}R$ and $0{\neq}abc{\in}I$, then $ab{\in}I$ or $ac{\in}\sqrt{I}$ or $bc{\in}\sqrt{I}$. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.
On weakly $S$-prime submodules
Hani A. Khashan,Ece Yetkin Celikel 대한수학회 2022 대한수학회보 Vol.59 No.6
Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ to be weakly $S$-prime if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in(N:_{R}M)$ or $sm\in N$. Many properties, examples and characterizations of weakly $S$-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly $S$-prime.
ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS
Ayman Badawi,Unsal Tekir,Ece Yetkin 대한수학회 2015 대한수학회지 Vol.52 No.1
Let R be a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, c ∈ R and 0 ≠ abc ∈ I, then ab ∈ I or ac ∈ √I or bc ∈ √I. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.
On graded $J$-ideals over graded rings
Tamem Al-Shorman,Malik Bataineh,Ece Yetkin Celikel 대한수학회 2023 대한수학회논문집 Vol.38 No.2
The goal of this article is to present the graded $J$-ideals of $G$-graded rings which are extensions of $J$-ideals of commutative rings. A graded ideal $P$ of a $G$-graded ring $R$ is a graded $J$-ideal if whenever $x,y\in h(R)$, if $xy\in P$ and $x\not\in J(R)$, then $y\in P$, where $h(R)$ and $J(R)$ denote the set of all homogeneous elements and the Jacobson radical of $R$, respectively. Several characterizations and properties with supporting examples of the concept of graded $J$-ideals of graded rings are investigated.