• Title/Summary/Keyword: weakly prime ideal

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THE WEAKLY SEMI-PRIME IDEALS OF po-Γ-SEMIGROUPS

  • Kwon, Young In;Lee, Sang Keun
    • Korean Journal of Mathematics
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    • v.5 no.2
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    • pp.135-139
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    • 1997
  • We introduce the concepts of weakly prime and weakly semi-prime ideals in po-${\Gamma}$-semigroup and give some characterizations of weakly prime and weakly semi-prime ideals of po-${\Gamma}$-semigroups analogous to the characterizations of weakly prime and weakly semi-prime ideals of po-semigroups considered by N. Kehayopulu.

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ON WEAKLY (m, n)-PRIME IDEALS OF COMMUTATIVE RINGS

  • Hani A. Khashan;Ece Yetkin Celikel
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.717-734
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    • 2024
  • Let R be a commutative ring with identity and m, n be positive integers. In this paper, we introduce the class of weakly (m, n)-prime ideals generalizing (m, n)-prime and weakly (m, n)-closed ideals. A proper ideal I of R is called weakly (m, n)-prime if for a, b ∈ R, 0 ≠ amb ∈ I implies either an ∈ I or b ∈ I. We justify several properties and characterizations of weakly (m, n)-prime ideals with many supporting examples. Furthermore, we investigate weakly (m, n)-prime ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behaviour of this class of ideals in idealization and amalgamated rings.

WEAKLY PRIME LEFT IDEALS IN NEAR-SUBTRACTION SEMIGROUPS

  • Dheena, P.;Kumar, G. Satheesh
    • Communications of the Korean Mathematical Society
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    • v.23 no.3
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    • pp.325-331
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    • 2008
  • In this paper we introduce the notion of weakly prime left ideals in near-subtraction semigroups. Equivalent conditions for a left ideal to be weakly prime are obtained. We have also shown that if (M, L) is a weak $m^*$-system and if P is a left ideal which is maximal with respect to containing L and not meeting M, then P is weakly prime.

WEAKLY PRIME IDEALS IN COMMUTATIVE SEMIGROUPS

  • Anderson, D.D.;Chun, Sangmin;Juett, Jason R.
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.829-839
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    • 2019
  • Let S be a commutative semigroup with 0 and 1. A proper ideal P of S is weakly prime if for $a,\;b{\in}S$, $0{\neq}ab{\in}P$ implies $a{\in}P$ or $b{\in}P$. We investigate weakly prime ideals and related ideals of S. We also relate weakly prime principal ideals to unique factorization in commutative semigroups.

INTUITIONISTIC FUZZY IDEALS OF A RING

  • Hur, Kul;Jang, Su-Youn;Kang, Hee-Won
    • The Pure and Applied Mathematics
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    • v.12 no.3 s.29
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    • pp.193-209
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    • 2005
  • We introduce the notions of intuitionistic fuzzy prime ideals, intuitionistic fuzzy completely prime ideals and intuitionistic fuzzy weakly completely prime ideals. And we give a characterization of intuitionistic fuzzy ideals and establish relationships between intuitionistic fuzzy completely prime ideals and intuitionistic fuzzy weakly completely prime ideals.

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Weakly Prime Ideals in Involution po-Γ-Semigroups

  • Abbasi, M.Y.;Basar, Abul
    • Kyungpook Mathematical Journal
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    • v.54 no.4
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    • pp.629-638
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    • 2014
  • The concept of prime and weakly prime ideal in semigroups has been introduced by G. Szasz [4]. In this paper, we define the involution in po-${\Gamma}$-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-${\Gamma}$-semigroup S. Also, we characterize intra-regular involution po-${\Gamma}$-semigroups. We establish that in the involution po-${\Gamma}$-semigroup S such that the involution preserves the order, an ideal of S is prime if and only if it is both weakly prime and semiprime and if S is commutative, then the prime and weakly prime ideals of S coincide. Finally, we prove that if S is a po-${\Gamma}$-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.

M-SYSTEM AND N-SYSTEM IN PO-SEMIGROUPS

  • Lee, Sang-Keun
    • East Asian mathematical journal
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    • v.19 no.2
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    • pp.233-240
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    • 2003
  • Xie and Wu introduced an m-system in a po-semigroup. Kehayopulu gave characterizations of weakly prime ideals of po-semigroups and Lee and Kwon add two characterizations for weakly prime ideals. In this paper, we give a characterization of weakly prime ideals and a characterization of weakly semi-prime ideals in po-semigroups using m-system and n-system, respectively

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ON WEAKLY PRIME IDEALS OF ORDERED ${\gamma}$-SEMIGROUPS

  • Kwon, Young-In;Lee, Sang-Keun
    • Communications of the Korean Mathematical Society
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    • v.13 no.2
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    • pp.251-256
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    • 1998
  • We introduce the concept of weakly prime ideals in po-$\Gamma$-semigroup and give some characterizations of weakly prime ideals.

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ON WEAKLY S-PRIME SUBMODULES

  • Hani A., Khashan;Ece Yetkin, Celikel
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.6
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    • pp.1387-1408
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    • 2022
  • Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :R M)∩S = ∅ to be weakly S-prime if there exists s ∈ S such that whenever a ∈ R and m ∈ M with 0 ≠ am ∈ N, then either sa ∈ (N :R M) or sm ∈ N. Many properties, examples and characterizations of weakly S-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly S-prime.

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.