• Title/Summary/Keyword: ${\delta}$-ideal

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PRIME RADICALS IN ORE EXTENSIONS

  • Han, Jun-Cheol
    • East Asian mathematical journal
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    • v.18 no.2
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    • pp.271-282
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    • 2002
  • Let R be a ring with an endomorphism $\sigma$ and a derivation $\delta$. An ideal I of R is ($\sigma,\;\delta$)-ideal of R if $\sigma(I){\subseteq}I$ and $\delta(I){\subseteq}I$. An ideal P of R is a ($\sigma,\;\delta$)-prime ideal of R if P(${\neq}R$) is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideals I and J of R, $IJ{\subseteq}P$ implies that $I{\subseteq}P$ or $J{\subseteq}P$. An ideal Q of R is ($\sigma,\;\delta$)-semiprime ideal of R if Q is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideal I of R, $I^2{\subseteq}Q$ implies that $I{\subseteq}Q$. The ($\sigma,\;\delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($\sigma,\;\delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(\sigma,\delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(\sigma,\delta)}(R)$ is the smallest ($\sigma,\;\delta$)-semiprime ideal of R; (2) For every extended endomorphism $\bar{\sigma}$ of $\sigma$, the $\bar{\sigma}$-prime radical of an Ore extension $P(R[x;\sigma,\delta])$ is equal to $P_{\sigma,\delta}(R)[x;\sigma,\delta]$.

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MORE GENERALIZED FUZZY SUBSEMIGROUPS/IDEALS IN SEMIGROUPS

  • Khan, Muhammad Sajjad Ali;Abdullah, Saleem;Jun, Young Bi;Rahman, Khaista
    • Honam Mathematical Journal
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    • v.39 no.4
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    • pp.527-559
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    • 2017
  • The main motivation of this article is to generalized the concept of fuzzy ideals, (${\alpha},{\beta}$)-fuzzy ideals, (${\in},{\in}{\vee}q_k$)-fuzzy ideals of semigroups. By using the concept of $q^{\delta}_K$-quasi-coincident of a fuzzy point with a fuzzy set, we introduce the notions of (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy left ideal, (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy right ideal of a semigroup. Special sets, so called $Q^{\delta}_k$-set and $[{\lambda}^{\delta}_k]_t$-set, condition for the $Q^{\delta}_k$-set and $[{\lambda}^{\delta}_k]_t$-set-set to be left (resp. right) ideals are considered. We finally characterize different classes of semigroups (regular, left weakly regular, right weakly regular) in term of (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy left ideal, (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy right ideal and (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy ideal of semigroup S.

CHARACTERIZATIONS OF IDEAL WEAKLY \delta\theta-REFINABLE SPACES

  • Cho, Myung-Hyun
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.33-45
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    • 1999
  • In this paper, we are interested in studying weak covering properties in the presence of a countable compact condition. The purpose of this paper is to characterize an ideal weakly $\delta$$\theta$-refinable space and to show that every ideal weakly $\delta$$\theta$-refinable space is isocompact. Also, we consider the behavior under mappings of ideal weakly $\delta$$\theta$-refinable properties and productivity of ideal weakly $\delta$$\theta$-refinable properties.

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𝛿;-FUZZY IDEALS IN PSEUDO-COMPLEMENTED DISTRIBUTIVE LATTICES

  • ALABA, BERHANU ASSAYE;NORAHUN, WONDWOSEN ZEMENE
    • Journal of applied mathematics & informatics
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    • v.37 no.5_6
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    • pp.383-397
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    • 2019
  • In this paper, we introduce ${\delta}$-fuzzy ideals in a pseudo complemented distributive lattice in terms of fuzzy filters. It is proved that the set of all ${\delta}$-fuzzy ideals forms a complete distributive lattice. The set of equivalent conditions are given for the class of all ${\delta}$-fuzzy ideals to be a sub-lattice of the fuzzy ideals of L. Moreover, ${\delta}$-fuzzy ideals are characterized in terms of fuzzy congruences.

2-absorbing δ-semiprimary Ideals of Commutative Rings

  • Celikel, Ece Yetkin
    • Kyungpook Mathematical Journal
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    • v.61 no.4
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    • pp.711-725
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    • 2021
  • Let R be a commutative ring with nonzero identity, 𝓘(𝓡) the set of all ideals of R and δ : 𝓘(𝓡) → 𝓘(𝓡) an expansion of ideals of R. In this paper, we introduce the concept of 2-absorbing δ-semiprimary ideals in commutative rings which is an extension of 2-absorbing ideals. A proper ideal I of R is called 2-absorbing δ-semiprimary ideal if whenever a, b, c ∈ R and abc ∈ I, then either ab ∈ δ(I) or bc ∈ δ(I) or ac ∈ δ(I). Many properties and characterizations of 2-absorbing δ-semiprimary ideals are obtained. Furthermore, 2-absorbing δ1-semiprimary avoidance theorem is proved.

FOOTPRINT AND MINIMUM DISTANCE FUNCTIONS

  • Nunez-Betancourt, Luis;Pitones, Yuriko;Villarreal, Rafael H.
    • Communications of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.85-101
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    • 2018
  • Let S be a polynomial ring over a field K, with a monomial order ${\prec}$, and let I be an unmixed graded ideal of S. In this paper we study two functions associated to I: The minimum distance function ${\delta}_I$ and the footprint function $fp_I$. It is shown that ${\delta}_I$ is positive and that $fp_I$ is positive if the initial ideal of I is unmixed. Then we show that if I is radical and its associated primes are generated by linear forms, then ${\delta}_I$ is strictly decreasing until it reaches the asymptotic value 1. If I is the edge ideal of a Cohen-Macaulay bipartite graph, we show that ${\delta}_I(d)=1$ for d greater than or equal to the regularity of S/I. For a graded ideal of dimension ${\geq}1$, whose initial ideal is a complete intersection, we give an exact sharp lower bound for the corresponding minimum distance function.

SQUAREFREE ZERO-DIVISOR GRAPHS OF STANLEY-REISNER RINGS

  • Nikseresht, Ashkan
    • Journal of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1381-1388
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    • 2018
  • Let ${\Delta}$ be a simplicial complex, $I_{\Delta}$ its Stanley-Reisner ideal and $K[{\Delta}]$ its Stanley-Reisner ring over a field K. Assume that ${\Gamma}(R)$ denotes the zero-divisor graph of a commutative ring R. Here, first we present a condition on two reduced Noetherian rings R and R', equivalent to ${\Gamma}(R){\cong}{\Gamma}(R{^{\prime}})$. In particular, we show that ${\Gamma}(K[{\Delta}]){\cong}{\Gamma}(K^{\prime}[{\Delta}^{\prime}])$ if and only if ${\mid}Ass(I_{\Delta}){\mid}={\mid}Ass(I_{{{\Delta}^{\prime}}}){\mid}$ and either ${\mid}K{\mid}$, ${\mid}K^{\prime}{\mid}{\leq}{\aleph}_0$ or ${\mid}K{\mid}={\mid}K^{\prime}{\mid}$. This shows that ${\Gamma}(K[{\Delta}])$ contains little information about $K[{\Delta}]$. Then, we define the squarefree zero-divisor graph of $K[{\Delta}]$, denoted by ${\Gamma}_{sf}(K[{\Delta}])$, and prove that ${\Gamma}_{sf}(K[{\Delta}){\cong}{\Gamma}_{sf}(K[{\Delta}^{\prime}])$ if and only if $K[{\Delta}]{\cong}K[{\Delta}^{\prime}]$. Moreover, we show how to find dim $K[{\Delta}]$ and ${\mid}Ass(K[{\Delta}]){\mid}$ from ${\Gamma}_{sf}(K[{\Delta}])$.

CONNECTEDNESS IN IDEAL PROXIMITY SPACES

  • Singh, Beenu;Singh, Davinder
    • Honam Mathematical Journal
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    • v.43 no.1
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    • pp.123-129
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    • 2021
  • Two new concepts, namely, ��∗-connectedness and ��∗-component are introduced by using ideal in proximity spaces. A relation of ��∗-connectedness with different types of connectedness that are considered in literature before is studied. It is shown that ��∗-connectedness is a contractive property.