• Title/Summary/Keyword: weakly primary ideal

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ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Badawi, Ayman;Tekir, Unsal;Yetkin, Ece
    • 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.

On 2-Absorbing and Weakly 2-Absorbing Primary Ideals of a Commutative Semiring

  • Soheilnia, Fatemeh
    • 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.

SOME RESULTS ON 1-ABSORBING PRIMARY AND WEAKLY 1-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Nikandish, Reza;Nikmehr, Mohammad Javad;Yassine, Ali
    • 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.

ON GRADED 2-ABSORBING PRIMARY AND GRADED WEAKLY 2-ABSORBING PRIMARY IDEALS

  • Al-Zoubi, Khaldoun;Sharafat, Nisreen
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.675-684
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    • 2017
  • Let G be a group with identity e and let R be a G-graded ring. In this paper, we introduce and study graded 2-absorbing primary and graded weakly 2-absorbing primary ideals of a graded ring which are different from 2-absorbing primary and weakly 2-absorbing primary ideals. We give some properties and characterizations of these ideals and their homogeneous components.

ON WEAKLY COMPLETELY QUASI PRIMARY AND COMPLETELY QUASI PRIMARY IDEALS IN TERNARY SEMIRINGS

  • Yiarayong, Pairote
    • Communications of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.657-665
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    • 2016
  • In this investigation we studied completely quasi primary and weakly completely quasi primary ideals in ternary semirings. Some characterizations of completely quasi primary and weakly completely quasi primary ideals were obtained. Moreover, we investigated relationships between completely quasi primary and weakly completely quasi primary ideals in ternary semirings. Finally, we obtained necessary and sufficient conditions for a weakly completely quasi primary ideal to be a completely quasi primary ideal.

(m, n)-CLOSED δ-PRIMARY IDEALS IN AMALGAMATION

  • Mohammad Hamoda;Mohammed Issoual
    • Communications of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.575-583
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    • 2024
  • Let R be a commutative ring with 1 ≠ 0. Let Id(R) be the set of all ideals of R and let δ : Id(R) → Id(R) be a function. Then δ is called an expansion function of the ideals of R if whenever L, I, J are ideals of R with J ⊆ I, then L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of the ideals of R and m ≥ n > 0 be positive integers. Then a proper ideal I of R is called an (m, n)-closed δ-primary ideal (resp., weakly (m, n)-closed δ-primary ideal ) if am ∈ I for some a ∈ R implies an ∈ δ(I) (resp., if 0 ≠ am ∈ I for some a ∈ R implies an ∈ δ(I)). Let f : A → B be a ring homomorphism and let J be an ideal of B. This paper investigates the concept of (m, n)-closed δ-primary ideals in the amalgamation of A with B along J with respect to f denoted by A ⋈f J.

SOME EXAMPLES OF WEAKLY FACTORIAL RINGS

  • Chang, Gyu Whan
    • Korean Journal of Mathematics
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    • v.21 no.3
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    • pp.319-323
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    • 2013
  • Let D be a principal ideal domain, X be an indeterminate over D, D[X] be the polynomial ring over D, and $R_n=D[X]/(X^n)$ for an integer $n{\geq}1$. Clearly, $R_n$ is a commutative Noetherian ring with identity, and hence each nonzero nonunit of $R_n$ can be written as a finite product of irreducible elements. In this paper, we show that every irreducible element of $R_n$ is a primary element, and thus every nonunit element of $R_n$ can be written as a finite product of primary elements.