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Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice

Received: 25 June 2024     Accepted: 18 July 2024     Published: 8 August 2024
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Abstract

Let ℜ be the finite commutative ring with unity and Is be the S-prime ideal of a ring ℜ. The set Ls forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (Ls, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.

Published in Applied and Computational Mathematics (Volume 13, Issue 4)
DOI 10.11648/j.acm.20241304.14
Page(s) 105-110
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Prime Ideal, S-prime Ideal, Nilpotent Ideal, Meet-semilattice

References
[1] Ahmed Hamed, Achraf Malek. S-prime ideals of a commutative ring. Beitrage zur Algebra and Geometrie.2020, 61, 533-542.
[2] Wala’a Alkasasbeh, Malik Bataineh. Generalization of S-prime ideals. Wseas Transactions on Mathematics. 2021, 20, 694-699.
[3] Hani A. Khashan, Amal B. Bani-Ata. J-ideals of commutative rings, International Electronic Journal of Algebra. 2021, 29, 148-164.
[4] Mohammed Tamekkante, El Mehdi Bouba. (2, n)-ideals of commutative rings. Journal of Algebra and Its Applications. 2019, 18(6), 1-12.
[5] Ali Akbar Estaji, Toktam Haghdadi, On n- absorbing Ideals in a Lattice. Kragujevac Journal of Mathematics. 2021, 45(4), 597-605.
[6] Meenakshi P. Wasadikar, Karuna T. Gaikwad. Some properties of 2-absorbing primary ideals in lattices. AKCE Int. J. Graphs Comb., 2019, 16, 18-26.
[7] Mihaela Istrata. Pure ideals in residuated lattices, Transactions on Fuzzy Sets and Systems. 2022, 1, 42-58.
[8] Wondwosen Zemene Norahun. O-Fuzzy ideals in distributivelattices. ResearchInMathematics. 2023, 10, 1-7.
[9] Alfred Horn, Naoki Kimura, The Category of Semilattices, Algebra Univ.. 1971, 26-38.
[10] Elliott Evans. The Boolean Ring Universal Over a Meet Semilattice, J. Austral.Math. Soc. 1977, 23, 402-415.
[11] K. Aiswarya, A. Afrinayesha. Semi Prime Filters in Meet Semilattice, International Research Journal of Engineering Technology. 2020, 7(2), 461-464.
[12] A. Afrin Ayesha, K. Aiswarya, 0-Distributive Meet-Semilattice, International Research Journal of Engineering and Technology. 2020, 7(2), 84-89.
[13] B. Davey, H. Priestley, Introduction to Lattices and Order. Second Edition. Cambridge University Press, 2002.
[14] D. Kalamani, C. V. Mythily, S-prime ideal graph of a finite commutative ring, Advances and Applications in Mathematical Sciences. 2023, 22 (4), 861-872.
[15] C. V. Mythily, D. Kalamani, Study on S-prime Ideal as Nilpotent Ideal, Journal of Applied Mathematics and Informatics, 2024.
[16] D. Kalamani, G. Ramya, Product Maximal Graph of a Finite Commutative Ring, Bull. Cal. Math. Soc. 2021, 113 (2), 127-134.
[17] G. Kiruthika, D. Kalamani, Degree based partition of the power graphs of a finite abelian group, Malaya Journal of Matematik. 2020, 1, 66-71.
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    Duraisamy, K., Varadharajan, M. (2024). Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice. Applied and Computational Mathematics, 13(4), 105-110. https://doi.org/10.11648/j.acm.20241304.14

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    ACS Style

    Duraisamy, K.; Varadharajan, M. Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice. Appl. Comput. Math. 2024, 13(4), 105-110. doi: 10.11648/j.acm.20241304.14

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    AMA Style

    Duraisamy K, Varadharajan M. Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice. Appl Comput Math. 2024;13(4):105-110. doi: 10.11648/j.acm.20241304.14

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  • @article{10.11648/j.acm.20241304.14,
      author = {Kalamani Duraisamy and Mythily Varadharajan},
      title = {Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice},
      journal = {Applied and Computational Mathematics},
      volume = {13},
      number = {4},
      pages = {105-110},
      doi = {10.11648/j.acm.20241304.14},
      url = {https://doi.org/10.11648/j.acm.20241304.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241304.14},
      abstract = {Let ℜ be the finite commutative ring with unity and Is be the S-prime ideal of a ring ℜ. The set Ls forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (Ls, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.},
     year = {2024}
    }
    

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    AU  - Kalamani Duraisamy
    AU  - Mythily Varadharajan
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    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    EP  - 110
    PB  - Science Publishing Group
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    AB  - Let ℜ be the finite commutative ring with unity and Is be the S-prime ideal of a ring ℜ. The set Ls forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (Ls, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.
    VL  - 13
    IS  - 4
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