• 제목/요약/키워드: Primary ideals

검색결과 45건 처리시간 0.02초

ON STRONGLY 1-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Almahdi, Fuad Ali Ahmed;Bouba, El Mehdi;Koam, Ali N.A.
    • 대한수학회보
    • /
    • 제57권5호
    • /
    • pp.1205-1213
    • /
    • 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.

On Graded 2-Absorbing and Graded Weakly 2-Absorbing Primary Ideals

  • Soheilnia, Fatemeh;Darani, Ahmad Yousefian
    • Kyungpook Mathematical Journal
    • /
    • 제57권4호
    • /
    • pp.559-580
    • /
    • 2017
  • Let G be an arbitrary group with identity e and let R be a G-graded ring. In this paper, we define the concept of graded 2-absorbing and graded weakly 2-absorbing primary ideals of commutative G-graded rings with non-zero identity. A number of results and basic properties of graded 2-absorbing primary and graded weakly 2-absorbing primary ideals are given.

ON STRONGLY QUASI PRIMARY IDEALS

  • Koc, Suat;Tekir, Unsal;Ulucak, Gulsen
    • 대한수학회보
    • /
    • 제56권3호
    • /
    • pp.729-743
    • /
    • 2019
  • In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if $ab{\in}Q$ for $a,b{\in}R$ implies either $a^2{\in}Q$ or $b^n{\in}Q$ ($a^n{\in}Q$ or $b^2{\in}Q$) for some $n{\in}{\mathbb{N}}$. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph ${\Gamma}_I(R)$ and denote it by ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.

On 2-absorbing Primary Ideals of Commutative Semigroups

  • Mandal, Manasi;Khanra, Biswaranjan
    • Kyungpook Mathematical Journal
    • /
    • 제62권3호
    • /
    • pp.425-436
    • /
    • 2022
  • In this paper we introduce the notion of 2-absorbing primary ideals of a commutative semigroup. We establish the relations between 2-absorbing primary ideals and prime, maximal, semiprimary and 2-absorbing ideals. We obtain various characterization theorems for commutative semigroups in which 2-absorbing primary ideals are prime, maximal, semiprimary and 2-absorbing ideals. We also study some other important properties of 2-absorbing primary ideals of a commutative semigroup.

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

  • Nikandish, Reza;Nikmehr, Mohammad Javad;Yassine, Ali
    • 대한수학회보
    • /
    • 제58권5호
    • /
    • 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 WEAKLY COMPLETELY QUASI PRIMARY AND COMPLETELY QUASI PRIMARY IDEALS IN TERNARY SEMIRINGS

  • Yiarayong, Pairote
    • 대한수학회논문집
    • /
    • 제31권4호
    • /
    • pp.657-665
    • /
    • 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.

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

  • Soheilnia, Fatemeh
    • Kyungpook Mathematical Journal
    • /
    • 제56권1호
    • /
    • 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
    • 대한수학회지
    • /
    • 제54권2호
    • /
    • 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.

ON 𝜙-n-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Mostafanasab, Hojjat;Darani, Ahmad Yousefian
    • 대한수학회지
    • /
    • 제53권3호
    • /
    • pp.549-582
    • /
    • 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.