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Galoitica: Journal of Mathematical Structures and Applications

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Galoitica: Journal of Mathematical Structures and Applications
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Volume 12 • Issue 1 • PP: 19-23 • 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 )

Lee Xu 1*
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,
Olalekan Joosati 2
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1University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China
2Cape Peninsula University of Technology, Faculty of Applied Science, South Africa
* Corresponding Author.
DOI https://doi.org/10.54216/GJMSA.0120103
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© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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

 

 

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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. Volume 12, no. Issue 1, 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, Volume 12(Issue 1), 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 Volume 12, no. Issue 1 (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, Volume 12(Issue 1), pp. 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. 2025;Volume 12(Issue 1):19-23. DOI: https://doi.org/10.54216/GJMSA.0120103
L. Xu, O. Joosati, "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. Volume 12, no. Issue 1, pp. 19-23, 2025. DOI: https://doi.org/10.54216/GJMSA.0120103
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