• Title/Summary/Keyword: weakly 2-absorbing ideal

Search Result 6, Processing Time 0.016 seconds

ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Badawi, Ayman;Tekir, Unsal;Yetkin, Ece
    • Journal of the Korean Mathematical Society
    • /
    • v.52 no.1
    • /
    • pp.97-111
    • /
    • 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
    • /
    • v.56 no.1
    • /
    • pp.107-120
    • /
    • 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.

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

  • Al-Zoubi, Khaldoun;Sharafat, Nisreen
    • Journal of the Korean Mathematical Society
    • /
    • v.54 no.2
    • /
    • pp.675-684
    • /
    • 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.

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
    • /
    • v.58 no.5
    • /
    • pp.1069-1078
    • /
    • 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 2-Absorbing and Weakly 2-Absorbing Ideals of Commutative Semirings

  • Darani, Ahmad Yousefian
    • Kyungpook Mathematical Journal
    • /
    • v.52 no.1
    • /
    • pp.91-97
    • /
    • 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 $R$.

WEAKLY (m, n)-CLOSED IDEALS AND (m, n)-VON NEUMANN REGULAR RINGS

  • Anderson, David F.;Badawi, Ayman;Fahid, Brahim
    • Journal of the Korean Mathematical Society
    • /
    • v.55 no.5
    • /
    • pp.1031-1043
    • /
    • 2018
  • Let R be a commutative ring with $1{\neq}0$, I a proper ideal of R, and m and n positive integers. In this paper, we define I to be a weakly (m, n)-closed ideal if $0{\neq}x^m\;{\in}I$ for $x{\in}R$ implies $x^n{\in}I$, and R to be an (m, n)-von Neumann regular ring if for every $x{\in}R$, there is an $r{\in}R$ such that $x^mr=x^n$. A number of results concerning weakly(m, n)-closed ideals and (m, n)-von Neumann regular rings are given.