Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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SOME REMARKS ON S-VALUATION DOMAINS

  • Ali Benhissi (Mathematics Department Faculty of Sciences of Monastir) ;
  • Abdelamir Dabbabi (Mathematics Department Faculty of Sciences of Monastir)
  • Received : 2023.05.08
  • Accepted : 2023.06.29
  • Published : 2024.01.31
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Abstract

Let A be a commutative integral domain with identity element and S a multiplicatively closed subset of A. In this paper, we introduce the concept of S-valuation domains as follows. The ring A is said to be an S-valuation domain if for every two ideals I and J of A, there exists s ∈ S such that either sI ⊆ J or sJ ⊆ I. We investigate some basic properties of S-valuation domains. Many examples and counterexamples are provided.

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Acknowledgement

The authors would like to thank the referee for his/her valuable comments which helps us improve the presentation of our paper.

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. A. Benhissi, Chain conditions in commutative rings, revised and translated from the French original, Springer, Cham, 2022. https://doi.org/10.1007/978-3-031-09898-7
  3. A. Benhissi and A. Dabbabi, An associated sequence of ideals of an increasing sequence of rings, Bull. Korean Math. Soc. 59 (2022), no. 6, 1349-1357. https://doi.org/10.4134/BKMS.b210425
  4. J. W. Brewer, Power series over commutative rings, Lecture Notes in Pure and Applied Mathematics, 64, Marcel Dekker, Inc., New York, 1981.
  5. A. Hamed, S-Noetherian spectrum condition, Comm. Algebra 46 (2018), no. 8, 3314-3321. https://doi.org/10.1080/00927872.2017.1412455
  6. A. Hamed and S. Hizem, S-Noetherian rings of the forms A[X] and A[[X]], Comm. Algebra 43 (2015), no. 9, 3848-3856. https://doi.org/10.1080/00927872.2014.924127