• Title/Summary/Keyword: amalgamated algebra along an ideal

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BÉZOUT RINGS AND WEAKLY BÉZOUT RINGS

  • El Alaoui, Haitham
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.4
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    • pp.843-852
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    • 2022
  • In this paper, we study some properties of Bézout and weakly Bézout rings. Then, we investigate the transfer of these notions to trivial ring extensions and amalgamated algebras along an ideal. Also, in the context of domains we show that the amalgamated is a Bézout ring if and only if it is a weakly Bézout ring. All along the paper, we put the new results to enrich the current literature with new families of examples of non-Bézout weakly Bézout rings.

ON S-MULTIPLICATION RINGS

  • Mohamed Chhiti;Soibri Moindze
    • Journal of the Korean Mathematical Society
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    • v.60 no.2
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    • pp.327-339
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    • 2023
  • Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. In this article we introduce a new class of ring, called S-multiplication rings which are S-versions of multiplication rings. An R-module M is said to be S-multiplication if for each submodule N of M, sN ⊆ JM ⊆ N for some s ∈ S and ideal J of R (see for instance [4, Definition 1]). An ideal I of R is called S-multiplication if I is an S-multiplication R-module. A commutative ring R is called an S-multiplication ring if each ideal of R is S-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and S-P IR. Moreover, we generalize some properties of multiplication rings to S-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.

ON ALMOST QUASI-COHERENT RINGS AND ALMOST VON NEUMANN RINGS

  • El Alaoui, Haitham;El Maalmi, Mourad;Mouanis, Hakima
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.5
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    • pp.1177-1190
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    • 2022
  • Let R be a commutative ring with identity. We call the ring R to be an almost quasi-coherent ring if for any finite set of elements α1, …, αp and a of R, there exists a positive integer m such that the ideals $\bigcap{_{i=1}^{p}}\;R{\alpha}^m_i$ and AnnRm) are finitely generated, and to be almost von Neumann regular rings if for any two elements a and b in R, there exists a positive integer n such that the ideal (αn, bn) is generated by an idempotent element. This paper establishes necessary and sufficient conditions for the Nagata's idealization and the amalgamated algebra to inherit these notions. Our results allow us to construct original examples of rings satisfying the above-mentioned properties.

ALMOST WEAKLY FINITE CONDUCTOR RINGS AND WEAKLY FINITE CONDUCTOR RINGS

  • Choulli, Hanan;Alaoui, Haitham El;Mouanis, Hakima
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.327-335
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    • 2022
  • Let R be a commutative ring with identity. We call the ring R to be an almost weakly finite conductor if for any two elements a and b in R, there exists a positive integer n such that anR ∩ bnR is finitely generated. In this article, we give some conditions for the trivial ring extensions and the amalgamated algebras to be almost weakly finite conductor rings. We investigate the transfer of these properties to trivial ring extensions and amalgamation of rings. Our results generate examples which enrich the current literature with new families of examples of nonfinite conductor weakly finite conductor rings.

S-COHERENT PROPERTY IN TRIVIAL EXTENSION AND IN AMALGAMATED DUPLICATION

  • Mohamed Chhiti;Salah Eddine Mahdou
    • Communications of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.705-714
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    • 2023
  • Bennis and El Hajoui have defined a (commutative unital) ring R to be S-coherent if each finitely generated ideal of R is a S-finitely presented R-module. Any coherent ring is an S-coherent ring. Several examples of S-coherent rings that are not coherent rings are obtained as byproducts of our study of the transfer of the S-coherent property to trivial ring extensions and amalgamated duplications.

GRADED PSEUDO-VALUATION RINGS

  • Fatima-Zahra Guissi;Hwankoo Kim;Najib Mahdou
    • Journal of the Korean Mathematical Society
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    • v.61 no.5
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    • pp.953-973
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    • 2024
  • Let R = ⊕α∈Γ Rα be a commutative ring graded by an arbitrary torsionless monoid Γ. A homogeneous prime ideal P of R is said to be strongly homogeneous prime if aP and bR are comparable for any homogeneous elements a, b of R. We will say that R is a graded pseudo-valuation ring (gr-PVR for short) if every homogeneous prime ideal of R is strongly homogeneous prime. In this paper, we introduce and study the graded version of the pseudo-valuation rings which is a generalization of the gr-pseudo-valuation domains in the context of arbitrary Γ-graded rings (with zero-divisors). We then study the possible transfer of this property to the graded trivial ring extension and the graded amalgamation. Our goal is to provide examples of new classes of Γ-graded rings that satisfy the above mentioned property.

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.

NONNIL-S-COHERENT RINGS

  • Najib Mahdou;El Houssaine Oubouhou
    • Communications of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.45-58
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    • 2024
  • Let R be a commutative ring with identity. If the nilpotent radical N il(R) of R is a divided prime ideal, then R is called a ϕ-ring. Let R be a ϕ-ring and S be a multiplicative subset of R. In this paper, we introduce and study the class of nonnil-S-coherent rings, i.e., the rings in which all finitely generated nonnil ideals are S-finitely presented. Also, we define the concept of ϕ-S-coherent rings. Among other results, we investigate the S-version of Chase's result and Chase Theorem characterization of nonnil-coherent rings. We next study the possible transfer of the nonnil-S-coherent ring property in the amalgamated algebra along an ideal and the trivial ring extension.