Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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STUDY ON S-PRIME IDEAL AS NILPOTENT IDEAL

  • C.V. MYTHILY (Department of Mathematics, Bharathiar University PG Extension and Research Centre) ;
  • D. KALAMANI (Department of Mathematics, Bharathiar University PG Extension and Research Centre)
  • Received : 2024.01.25
  • Accepted : 2024.05.27
  • Published : 2024.09.30
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Abstract

Let S be a multiplicative subset of a commutative ring 𝓡 with unity and Is be an S-prime ideal of 𝓡 which is disjoint from the multiplicative subset S. In this paper, some properties of the S-prime ideal, namely sum, union and intersection of two S-prime ideals are studied in a commutative ring 𝓡 with unity. It is proved that a nilradical of 𝓡 is the S-prime ideal of 𝓡. Zorn's lemma is used to state that an S-prime ideal is unique in a local ring 𝓡. Finally, the S-prime ideals in the semilocal ring are classified. The generalized S-prime ideal and its multiplicative subsets of a finite commutative ring with unity are presented.

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References

  1. Ahmed Hamed and Achraf Malek, S-prime ideals of a commutative ring, Beitrage zur Algebra und Geometrie, 2019.
  2. I. Akray, I-prime ideals, Journal of Algebra and Related Topics 4 (2016), 41-47.
  3. D.D. Anderson and E. Smith, Weakly prime ideals, Houston Journal of Mathematics 29 (2003), 831-838.
  4. M. Atiyah and I.G. McDonald, Introduction to Commutative Algebra, Addison Wesley Publishing Company, 2018.
  5. A. Badawi, On 2-absorbing ideals of commutative rings, Bulletin of the Australian Mathematical Society 7 (2007), 417-429.
  6. Batool Zarei Jalal Abadi and Hosen Fazaeli Moghimi, On (m, n)-absorbing ideals of commutative rings, Proc. Indian Acad. Sci.(Math. Sci.) 127 (2017), 251-261.
  7. C. Beddani and W. Messirdi, 2-Prime ideals and their applications, Journal of Algebra and Its Applications 15 (2016), 1-11.
  8. B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, second ed., Cambridge University Press, 2002.
  9. David S. Dummit and R.M. Foote, Abstract Algebra, third edition., John Wiley and Sons Inc, 2004.
  10. Driss Bennis and Brahim Fahid, Rings in which every 2-absorbing ideal is prime, Beitr Algebra Geom. 59 (2017), 391-396.
  11. Fatemeh Soheilnia, On 2-absorbing and weakly 2-absorbing primary ideals of a commutative semiring, KYUNGPOOK Math. J. 56 (2016), 107-120.
  12. Fuad Ali Ahmahdi, El Mehdi Bouba and Mohammed Tamekkante, On weakly S- prime ideals of commutative rings, Analele Universitatii Ovidius Constanta- Seria Matematica 29 (2021), 173-186.
  13. Govindarajulu Narayanan Sudharshana, On weakly S-2-Absorbing submodules, Int. J. Anal. Appl. 21 (2023), 1-14.
  14. D. Kalamani and C.V. Mythily, S-prime ideal graph of a finite commutative ring, Advances and Applications in Mathematical Sciences 22 (2023), 861-872.
  15. D. Kalamani and G. Ramya, Product Maximal Graph of a Finite Commutative Ring, Bull. Cal. Math. Soc. 113 (2021), 127-134.
  16. D. Kalamani and G. Ramya, Graph theoretic properties for Γpm(R) and resistance distance based indices, Advances and Applications in Mathematical Sciences 21 (2022), 3213-3231.
  17. G. Kiruthika and D. Kalamani, Some aspects of Vertex- order graph, Italian Journal of Pure and Applied Mathematics 50 (2023), 374-389.
  18. Manish Kent Dubey, Prime and weakly prime ideals in semirings, Quasigroups and Related Systems 20 (2012), 197-202.
  19. Mohamed Aqalmoun, S-prime ideals in principal domain, J. Indones. Math. Soc. 29 (2023), 93-98.
  20. Suat Koc, On weakly 2-prime ideals in commutative ring, Communications in Algebra 49 (2021), 3387-3397.
  21. Wala'a Alkasasbeh and Malik Bataineh, Generalization of S-prime ideals, Wseas Transactions on Mathematics 20 (2021), 694-699.
  22. A. Yassine, M.J. Nikmehr and R. Nikandish, On 1-Absorbing prime ideals of commutative rings, Journal of Algebra and its Applications 20 (2020), 1-13.