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Galoitica: Journal of Mathematical Structures and Applications

Volume 12 , Issue 1 , PP: 19-23, 2025 | Cite this article as | XML | Html | PDF | Full Length Article

A Short Contribution to the Classification of the Group of Units of the Rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) ) and NCRZ_(p^2 )

Lee Xu 1 * , Olalekan Joosati 2

  • 1 University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China - (Leexu1244@yahoo.com)
  • 2 Cape Peninsula University of Technology, Faculty of Applied Science, South Africa - (Olalekanjoo1997@cput.ac.za)

    • Doi: https://doi.org/10.54216/GJMSA.0120103

    Received: October 04, 2024 Revised: December 08, 2024 Accepted: January 03, 2025
    Abstract

    In this paper, we study the group of units problem of three different non-commutative logical extensions rings, where we classify the group of units of the rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) )and (NCR)_(Z_(p^2 ) )as semi direct products of well-known abelian groups as the following:

    U(N⊂R)_(Z_pq )≅(Z_(p-1)×Z_(q-1) )∝[(Z_p×Z_q )∝(Z_(p-1)×Z_(q-1)),

    U(NCR)_(Z_(2^n ) )≅(Z_2×Z_(2^(n-2)))∝(Z_(2^n )∝(Z_2×Z_(2^(n-2)))),

     

    U(N⊂R)_(Z_(p^2 ) )≅Z_(p^2-p)∝(Z_(p^2 )∝Z_(p^2-p)).

    Keywords :

    Non-commutative logical extension , Group of units , Semi-direct product , Abelian subgroup

    References

    [1]    Terazic, S., and Podobnic, S. (2024). "On the Group of Units Problem of the Non-Commutative Logical Extension of the Rings 𝒁𝒑 and 𝒁𝟐𝒏." Galoitica: Journal of Mathematical Structures and Applications, vol. 12, no. 1, pp. 37-43. DOI: 10.54216/GJMSA.0110205.

    [2]    Ozcek, M. (2024). "On the Non-Commutative Logical Rings As Novel Extensions of Neutrosophic Rings." Journal of Neutrosophic and Fuzzy Systems, vol. 22, no. 3, pp. 23-30. DOI: 10.54216/JNFS.080203.

    [3]    Sankari, H., and Abobala, M. (2023). "On the Group of Units Classification In 3-Cyclic and 4-Cyclic Refined Rings of Integers and the Proof of Von Shtawzens' Conjectures." International Journal of Neutrosophic Science, vol. 15, no. 2, pp. 101-110. DOI: 10.54216/IJNS.030202.

    [4]    Basheer, A., Ahmad, K., and Ali, R. (2022). "A Short Contribution to Von Shtawzen's Abelian Group In n-Cyclic Refined Neutrosophic Rings." Journal of Neutrosophic and Fuzzy Systems, vol. 21, no. 4, pp. 15-22. DOI: 10.54216/JNFS.070204.

    [5]    Osagie, A. (2024). "On The Algebraic Classification of the 4-Cyclic Refined Neutrosophic Real Roots of Unity Group." Galoitica: Journal of Mathematical Structures and Applications, vol. 12, no. 2, pp. 28-36. DOI: 10.54216/GJMSA.0110204.

     

     

    Cite This Article As :
    Xu, Lee. , Joosati, Olalekan. A Short Contribution to the Classification of the Group of Units of the Rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) ) and NCRZ_(p^2 ). Galoitica: Journal of Mathematical Structures and Applications, vol. , no. , 2025, pp. 19-23. DOI: https://doi.org/10.54216/GJMSA.0120103
    Xu, L. Joosati, O. (2025). A Short Contribution to the Classification of the Group of Units of the Rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) ) and NCRZ_(p^2 ). Galoitica: Journal of Mathematical Structures and Applications, (), 19-23. DOI: https://doi.org/10.54216/GJMSA.0120103
    Xu, Lee. Joosati, Olalekan. A Short Contribution to the Classification of the Group of Units of the Rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) ) and NCRZ_(p^2 ). Galoitica: Journal of Mathematical Structures and Applications , no. (2025): 19-23. DOI: https://doi.org/10.54216/GJMSA.0120103
    Xu, L. , Joosati, O. (2025) . A Short Contribution to the Classification of the Group of Units of the Rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) ) and NCRZ_(p^2 ). Galoitica: Journal of Mathematical Structures and Applications , () , 19-23 . DOI: https://doi.org/10.54216/GJMSA.0120103
    Xu L. , Joosati O. [2025]. A Short Contribution to the Classification of the Group of Units of the Rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) ) and NCRZ_(p^2 ). Galoitica: Journal of Mathematical Structures and Applications. (): 19-23. DOI: https://doi.org/10.54216/GJMSA.0120103
    Xu, L. Joosati, O. "A Short Contribution to the Classification of the Group of Units of the Rings (NCR)_(Z_pq ), (NCR)_(Z_(2^n ) ) and NCRZ_(p^2 )," Galoitica: Journal of Mathematical Structures and Applications, vol. , no. , pp. 19-23, 2025. DOI: https://doi.org/10.54216/GJMSA.0120103

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