• Title/Summary/Keyword: (closed) ideal

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CONSTRUCTION OF QUOTIENT BCI(BCK)-ALGEBRA VIA A FUZZY IDEAL

  • Liu, Yong-Lin;Jie Meng
    • Journal of applied mathematics & informatics
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    • v.10 no.1_2
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    • pp.51-62
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    • 2002
  • The present paper gives a new construction of a quotient BCI(BCK)-algebra X/${\mu}$ by a fuzzy ideal ${\mu}$ in X and establishes the Fuzzy Homomorphism Fundamental Theorem. We show that if ${\mu}$ is a fuzzy ideal (closed fuzzy ideal) of X, then X/${\mu}$ is a commutative (resp. positive implicative, implicative) BCK(BCI)-algebra if and only if It is a fuzzy commutative (resp. positive implicative, implicative) ideal of X Moreover we prove that a fuzzy ideal of a BCI-algebra is closed if and only if it is a fuzzy subalgebra of X We show that if the period of every element in a BCI-algebra X is finite, then any fuzzy ideal of X is closed. Especiatly, in a well (resp. finite, associative, quasi-associative, simple) BCI-algebra, any fuzzy ideal must be closed.

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
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    • v.55 no.5
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    • pp.1031-1043
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    • 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.

GENERALIZED CLOSED SETS IN BINARY IDEAL TOPOLOGICAL SPACES

  • Modak, Shyamapada;Al-omari, Ahmad Abdullah
    • Journal of the Chungcheong Mathematical Society
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    • v.31 no.1
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    • pp.183-191
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    • 2018
  • This paper deals with binary ideal topological space and discuss about generalized binary closed sets and generalized kernel in the same topological space. Further it will discuss various types of characterizations of generalized binary closed sets and generalized kernel.

(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.

UNION-SOFT SETS WITH APPLICATIONS IN BCK/BCI-ALGEBRAS

  • Jun, Young Bae
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1937-1956
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    • 2013
  • The aim of this article is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of union-soft sets is introduced, and its application to BCK/BCI-algebras is considered. The notions of union-soft algebras, union-soft (commutative) ideals and closed union-soft ideals are introduced, and related properties and relations are investigated. Conditions for a union-soft ideal to be closed are provided. Conditions for a union-soft ideal to be a union-soft commutative ideal are also provided. Characterizations of (closed) union-soft ideals and union-soft commutative ideals are established. Extension property for a union-soft commutative ideal is established.

FALLING FUZZY BCI-COMMUTATIVE IDEALS

  • Jun, Young Bae;Song, Seok-Zun
    • Honam Mathematical Journal
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    • v.36 no.3
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    • pp.555-568
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    • 2014
  • On the basis of the theory of a falling shadow and fuzzy sets, the notion of a falling fuzzy BCI-commutative ideal of a BCI-algebra is introduced. Relations between falling fuzzy BCI-commutative ideals and falling fuzzy ideals are given. Relations between fuzzy BCI-commutative ideals and falling fuzzy BCI-commutative ideals are provided. Characterizations of a falling fuzzy BCI-commutative ideal are established, and conditions for a falling fuzzy (closed) ideal to be a falling fuzzy BCI-commutative ideal are considered.

k-NIL RADICAL IN BCI-ALGEBRAS II

  • Jun, Y.B;Hong, S.M
    • Communications of the Korean Mathematical Society
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    • v.12 no.3
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    • pp.499-505
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    • 1997
  • This paper is a continuation of [3]. We prove that if A is quasi-associative (resp. an implicative) ideal of a BCI-algebra X then the k-nil radical of A is a quasi-associative (resp. an implicative) ideal of X. We also construct the quotient algebra $X/[Z;k]$ of a BCI-algebra X by the k-nhil radical [A;k], and show that if A and B are closed ideals of BCI-algebras X and Y respectively, then

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$\mathcal I$-IDEALS GENERATED BY A SET IN IS-ALGEBRAS

ON THE PRINCIPAL IDEAL THEOREM

  • Chang, Gyu-Whan
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.4
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    • pp.655-660
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    • 1999
  • Let R be an integral domain with identity. In this paper we will show that if R is integrally closed or if t-dim $R{\leq}1$, then R[{$X_{\alpha}$}] satisfies the principal ideal theorem for each family {$X_{\alpha}$} of algebraically independent indeterminates if and only if R is an S-domain and it satisfies the principal ideal theorem.

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