• 대한수학회보
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• 제34권1호
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• pp.35-42
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• 1997

Throughout this paper, all rings are assumed to be commutative with identity. By a local ring (R,m), we mean a Noetherian ring R which has a unique maximal ideal m. Let I be an ideal in a ring R and t an indeterminate over R.

• Journal of applied mathematics & informatics
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• 제26권3_4호
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• pp.707-720
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• 2008

Molodtsov [12] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to commutative ideals of BCK-algebras, The notions of commutative soft ideals and commutative idealistic soft BCK-algebras are introduced, and their basic properties are investigated. Examples to show that there is no relations between positive implicative idealistic soft BCK-algebras and commutative idealistic soft BCK-algebras are provided.

• 대한수학회논문집
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• 제35권4호
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• pp.1045-1056
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• 2020

Let R be a commutative ring with unity. The extension of annihilating-ideal graph of R, $^{\bar{\mathbb{AG}}}$(R), is the graph whose vertices are nonzero annihilating ideals of R and two distinct vertices I and J are adjacent if and only if there exist n, m ∈ ℕ such that IⁿJ^m = (0) with Iⁿ, J^m ≠ (0). First, we differentiate when 𝔸𝔾(R) and $^{\bar{\mathbb{AG}}}$(R) coincide. Then, we have characterized the diameter and the girth of $^{\bar{\mathbb{AG}}}$(R) when R is a finite direct products of rings. Moreover, we show that $^{\bar{\mathbb{AG}}}$(R) contains a cycle, if $^{\bar{\mathbb{AG}}}$(R) ≠ 𝔸𝔾(R).

• 대한수학회보
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• 제57권1호
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• pp.117-126
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• 2020

Let R be a commutative ring with identity. A proper ideal I of R is called 2-prime if for all a, b ∈ R such that ab ∈ I, then either a² or b² lies in I. In this paper, we study 2-prime ideals which are generalization of prime ideals. Our study provides an analogous to the prime avoidance theorem and some applications of this theorem. Also, it is shown that if R is a PID, then the families of primary ideals and 2-prime ideals of R are identical. Moreover, a number of examples concerning 2-prime ideals are given. Finally, rings in which every 2-prime ideal is a prime ideal are investigated.

• Kyungpook Mathematical Journal
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• 제52권1호
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• pp.91-97
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• 2012

Let $R$ be a commutative semiring. We define a proper ideal $I$ of $R$ to be 2-absorbing (resp., weakly 2-absorbing) if $abc{\in}I$ (resp., $0{\neq}abc{\in}I$) implies $ab{\in}I$ or $ac{\in}I$ or $bc{\in}I$. We show that a weakly 2-absorbing ideal $I$ with $I^3{\neq}0$ is 2-absorbing. We give a number of results concerning 2-absorbing and weakly 2-absorbing ideals and examples of weakly 2-absorbing ideals. Finally we de ne the concept of 0 - (1-, 2-, 3-)2-absorbing ideals of $R$ and study the relationship among these classes of ideals of $R$.

• East Asian mathematical journal
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• 제9권1호
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• pp.1-6
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• 1993

• 충청수학회지
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• 제12권1호
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• pp.15-21
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• 1999

In this paper, we introduce the notion of anti fuzzy prime ideals in a commutative BCK-algebra and obtain some properties of it.

• Kyungpook Mathematical Journal
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• 제56권1호
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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.

• 대한수학회보
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• 제58권5호
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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.

• 한국지능시스템학회논문지
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• 제20권5호
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• pp.734-740
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• 2010

We define and study the fuzzy nil radical of a fuzzy ideal of a commutative $\kappa$-semiring.