• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1205-1213
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• 2020

Let R be a commutative ring with 1 ≠ 0. In this paper, we introduce a subclass of the class of 1-absorbing primary ideals called the class of strongly 1-absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ ${\sqrt{0}}$. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the UN-rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1069-1078
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• 2021

Let R be a commutative ring with identity. A proper ideal I of R is called 1-absorbing primary ([4]) if for all nonunit a, b, c ∈ R such that abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. The concept of 1-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly 1-absorbing primary ideals which are generalization of weakly prime ideals and 1-absorbing primary ideals. A proper ideal I of R is called weakly 1-absorbing primary if for all nonunit a, b, c ∈ R such that 0 ≠ abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. Some properties of weakly 1-absorbing primary ideals are investigated. For instance, weakly 1-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if I is a weakly 1-absorbing primary ideal of a ring R and 0 ≠ I₁I₂I₃ ⊆ I for some ideals I₁, I₂, I₃ of R such that I is free triple-zero with respect to I₁I₂I₃, then I₁I₂ ⊆ I or I₃ ⊆ I.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.107-120
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• 2016

Let R be a commutative semiring. The purpose of this note is to investigate the concept of 2-absorbing (resp., weakly 2-absorbing) primary ideals generalizing of 2-absorbing (resp., weakly 2-absorbing) ideals of semirings. A proper ideal I of R said to be a 2-absorbing (resp., weakly 2-absorbing) primary ideal if whenever $a,b,c{\in}R$ such that $abc{\in}I$ (resp., $0{\neq}abc{\in}I$), then either $ab{\in}I$ or $bc{\in}\sqrt{I}$ or $ac{\in}\sqrt{I}$. Moreover, when I is a Q-ideal and P is a k-ideal of R/I with $I{\subseteq}P$, it is shown that if P is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R, then P/I is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R/I and it is also proved that if I and P/I are weakly 2-absorbing primary ideals, then P is a weakly 2-absorbing primary ideal of R.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.549-582
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• 2016

All rings are commutative with $1{\neq}0$ and n is a positive integer. Let ${\phi}:{\Im}(R){\rightarrow}{\Im}(R){\cup}\{{\emptyset}\}$ be a function where ${\Im}(R)$ denotes the set of all ideals of R. We say that a proper ideal I of R is ${\phi}$-n-absorbing primary if whenever $a_1,a_2,{\cdots},a_{n+1}{\in}R$ and $a_1,a_2,{\cdots},a_{n+1}{\in}I{\backslash}{\phi}(I)$, either $a_1,a_2,{\cdots},a_n{\in}I$ or the product of $a_{n+1}$ with (n-1) of $a_1,{\cdots},a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of ${\phi}$-n-absorbing primary ideals.

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.97-111
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• 2015

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.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.711-725
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• 2021

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 δ₁-semiprimary avoidance theorem is proved.

• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1163-1173
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• 2014

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.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1505-1519
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• 2017

Assume that M is an R-module where R is a commutative ring. A proper submodule N of M is called a weakly 2-absorbing primary submodule of M if $0{\neq}abm{\in}N$ for any $a,b{\in}R$ and $m{\in}M$, then $ab{\in}(N:M)$ or $am{\in}M-rad(N)$ or $bm{\in}M-rad(N)$. In this paper, we extended the concept of weakly 2-absorbing primary ideals of commutative rings to weakly 2-absorbing primary submodules of modules. Among many results, we show that if N is a weakly 2-absorbing primary submodule of M and it satisfies certain condition $0{\neq}I_1I_2K{\subseteq}N$ for some ideals $I_1$, $I_2$ of R and submodule K of M, then $I_1I_2{\subseteq}(N:M)$ or $I_1K{\subseteq}M-rad(N)$ or $I_2K{\subseteq}M-rad(N)$.