Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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S-Q-IDEALS AND ALMOST S-Q-IDEALS OF COMMUTATIVE RINGS

  • Sihem Smach (Department of Mathematics Faculty of Sciences University of Monastir)
  • Received : 2024.10.29
  • Accepted : 2025.03.11
  • Published : 2025.10.31
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Abstract

Let R be a commutative ring with identity and S a multiplicative subset of R. This article is motivated by the results of Mimouni on Q-ideals. In this paper, we introduce and investigate the notion of S-Q-ideals which is a generalization of Q-ideals introduced by Mimouni in [8]. Let I ⊆ Q be two ideals of R with Q ∩ S = Ø. The ideal I is said to be an S-Q-ideal if there exists s ∈ S such that whenever ab ∈ I with a, b ∈ R, then sa ∈ I or sb ∈ Q. In Section 2, we study some properties of S-Q-ideals. Among other results, we give some characterization of the S-Q-ideal of the idealization of a module. On the other hand, we show I is an S-Q-ideal of R if and only if I[X] is an S-Q[X]-ideal of R[X], provided that R is a domain and S is anti-Archimedean. In Section 3, we introduce a weak version of S-Q-ideals and we study some of their properties.

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References

  1. D. D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra 30 (2002), no. 9, 4407–4416. https://doi.org/10.1081/AGB-120013328
  2. 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
  3. A. Dabbabi and A. Benhissi, Generalization of the S-Noetherian concept, Arch. Math. (Brno) 59 (2023), no. 4, 307–314. https://doi.org/10.5817/am2023-4-307
  4. R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972.
  5. A. Hamed and A. Malek, S-prime ideals of a commutative ring, Beitr. Algebra Geom. 61 (2020), no. 3, 533–542. https://doi.org/10.1007/s13366-019-00476-5
  6. H. A. Khashan and E. Yetkin Çelikel, Quasi J-ideals of commutative rings, Ric. Mat. 73 (2024), no. 4, 2035–2047. https://doi.org/10.1007/s11587-022-00716-2
  7. E. Massaoud, S-primary ideals of a commutative ring, Comm. Algebra 50 (2022), no. 3, 988–997. https://doi.org/10.1080/00927872.2021.1977939
  8. A. Mimouni, Nil-ideals, J-ideals and their generalizations in commutative rings, Beitr. Algebra Geom. 64 (2023), no. 4, 1043–1056. https://doi.org/10.1007/s13366-022-00668-6
  9. S. Visweswaran, Some results on S-primary ideals of a commatatioe ring, Beitr. Algebra Geom. 63 (2022), no. 2, 247-266. https://doi.org/10.1007/s13366-021-00580-5