• Title/Summary/Keyword: amalgamated duplication of a ring along an ideal

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ZPI Property In Amalgamated Duplication Ring

  • Hamed, Ahmed;Malek, Achraf
    • Kyungpook Mathematical Journal
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    • v.62 no.2
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    • pp.205-211
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    • 2022
  • Let A be a commutative ring. We say that A is a ZPI ring if every proper ideal of A is a finite product of prime ideals [5]. In this paper, we study when the amalgamated duplication of A along an ideal I, A ⋈ I to be a ZPI ring. We show that if I is an idempotent ideal of A, then A is a ZPI ring if and only if A ⋈ I is a ZPI ring.

SOME COMMUTATIVE RINGS DEFINED BY MULTIPLICATION LIKE-CONDITIONS

  • Chhiti, Mohamed;Moindze, Soibri
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.2
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    • pp.397-405
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    • 2022
  • In this article we investigate the transfer of multiplication-like properties to homomorphic images, direct products and amalgamated duplication of a ring along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

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.

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.

AMALGAMATED MODULES ALONG AN IDEAL

  • El Khalfaoui, Rachida;Mahdou, Najib;Sahandi, Parviz;Shirmohammadi, Nematollah
    • Communications of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.1-10
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    • 2021
  • Let R and S be two commutative rings, J be an ideal of S and f : R → S be a ring homomorphism. The amalgamation of R and S along J with respect to f, denoted by R ⋈f J, is the special subring of R × S defined by R ⋈f J = {(a, f(a) + j) | a ∈ R, j ∈ J}. In this paper, we study some basic properties of a special kind of R ⋈f J-modules, called the amalgamation of M and N along J with respect to ��, and defined by M ⋈�� JN := {(m, ��(m) + n) | m ∈ M and n ∈ JN}, where �� : M → N is an R-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.

S-NOETHERIAN IN BI-AMALGAMATIONS

  • Kim, Hwankoo;Mahdou, Najib;Zahir, Youssef
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.1021-1029
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    • 2021
  • This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the S-Noetherian property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. Our results allow us to construct new and original examples of rings satisfying the S-Noetherian property.