Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON S-MULTIPLICATION RINGS

  • Mohamed Chhiti (Modelling and Mathematical Structures Laboratory Faculty of Economics and Social Sciences of Fez University S.M. Ben Abdellah) ;
  • Soibri Moindze (Modelling and Mathematical Structures Laboratory Department of Mathematics Faculty of Science and Technology of Fez, University S.M. Ben Abdellah)
  • Received : 2022.01.18
  • Accepted : 2022.12.12
  • Published : 2023.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

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.

Keywords

References

  1. M. M. Ali, Multiplication modules and homogeneous idealization, Beitrage Algebra Geom. 47 (2006), no. 1, 249-270.
  2. M. M. Ali, Idealization and theorems of D. D. Anderson, Comm. Algebra 34 (2006), no. 12, 4479-4501. https://doi.org/10.1080/00927870600938837
  3. D. D. Anderson, Multiplication ideals, multiplication rings, and the ring R(X), Canadian J. Math. 28 (1976), no. 4, 760-768. https://doi.org/10.4153/CJM-1976-072-1
  4. D. D. Anderson, T. Arabaci, U. Tekir, and S. Ko,c, On S-multiplication modules, Comm. Algebra 48 (2020), no. 8, 3398-3407. https://doi.org/10.1080/00927872.2020.1737873
  5. D. D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra 30 (2002), no. 9, 4407-4416. https://doi.org/10.1081/AGB-120013328
  6. D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
  7. D. Bennis and M. El Hajoui, On S-coherence, J. Korean Math. Soc. 55 (2018), no. 6, 1499-1512. https://doi.org/10.4134/JKMS.j170797
  8. M. Chhiti and S. Moindze, Some commutative rings defined by multiplication likeconditions, Bull. Korean Math. Soc. 59 (2022), no. 2, 397-405. https://doi.org/10.4134/BKMS.b210293
  9. M. D'Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal, in Commutative algebra and its applications, 155-172, Walter de Gruyter, Berlin, 2009.
  10. M. D'Anna, C. A. Finocchiaro, and M. Fontana, New algebraic properties of an amalgamated algebra along an ideal, Comm. Algebra 44 (2016), no. 5, 1836-1851. https://doi.org/10.1080/00927872.2015.1033628
  11. M. D'Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (2007), no. 3, 443-459. https://doi.org/10.1142/ S0219498807002326
  12. L. Fuchs, Uber die Ideale arithmetischer Ringe, Comment. Math. Helv. 23 (1949), 334-341. https://doi.org/10.1007/BF02565607
  13. C. U. Jensen, Arithmetical rings, Acta Math. Acad. Sci. Hungar. 17 (1966), 115-123. https://doi.org/10.1007/BF02020446
  14. M. Kabbour and N. Mahdou, Arithmetical property in amalgamated algebras along an ideal, Palest. J. Math. 3 (2014), Special issue, 395-399.
  15. K. Louartiti and N. Mahdou, Transfer of multiplication-like conditions in amalgamated algebra along an ideal, Afr. Diaspora J. Math. 14 (2012), no. 1, 119-125.
  16. A. Mimouni, M. Kabbour, and N. Mahdou, Trivial ring extensions defined by arithmetical-like properties, Comm. Algebra 41 (2013), no. 12, 4534-4548. https://doi.org/10.1080/00927872.2012.705932
  17. M. Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York, 1962.