Homoderivations in Prime Near Rings
DOI:
https://doi.org/10.20956/zsw81k91Keywords:
homoderivation, prime near ring, commutativity theorem, Lie structureAbstract
Homoderivations represent a generalized differential operator that combines multiplicative and derivational properties, making them significant in studying algebraic structures with hybrid operations. This paper initiates the systematic study of homoderivations on prime near rings, an algebraic structure where only left distributivity holds. We establish the foundational Leibniz formula and investigate when the existence of such mapping forces the underlying near ring to become commutative. Specificall, we prove that if a 2-torsion free prime near ring admits a non zero homoderivation h with h([x,y])=0 or h(x◦y)=0, then N is a commutative ring. Furthermore, we characterize the center of the near ring through the behavior of homoderivations on square closed Lie ideals. These results have implications for understanding the interplay between ring theoritic derivation and multiplicative mappings in non ring.
References
[1] Alharfie, E.F., Muthana, N.M., & Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia, 2018. The commutativity of prime rings with homoderivations. International Journal of ADVANCED AND APPLIED SCIENCES. Vol. 5, No. 5, 79–81.
[2] Ashraf, M. & Ali, S., 2024. On (σ,τ)-derivations of Prime Near-Rings-II. Sarajevo Journal of Mathematics. Vol. 4, No. 1, 23–30.
[3] Belkadi, S., Ali, S. & Taoufiq, L., 2024. On n -Jordan homoderivations in rings. Georgian Mathematical Journal. Vol. 31, No. 1, 17–24.
[4] Bell, H.E. & Mason, G. 1987. On derivations in near-rings. North-Holland Mathematics Studies. Elsevier,
[5] Boua, A., Golbasi, O. & Mouhssine, S., 2024. Lie right ideals and homoderivations in 3-Prime Near-Ring. Boletim da Sociedade Paranaense de Matemática. Vol. 42, 1–12.
[6] El Amrani, S. & Raji, A., 2026. A study of homoderivations in 3-prime near-rings with centralizing constraints. Algebra and Discrete Mathematics. Vol. 40, No. 2
[7] El Sofy, M., 2000. Rings with some kinds of mappings. Cairo University, Branch of Fayoum (M. Sc. Thesis)
[8] El-Soufi, M.M. & Ghareeb, A., 2022. Centrally Extended α ‐Homoderivations on Prime and Semiprime Rings. Journal of Mathematics. Vol. 2022, No. 1, 2584177.
[9] Engin, A. & Aydın, N., 2023. Homoderivations in Prime Rings. Journal of New Theory, No. 43, 23–34.
[10] Hummdi, A.Y., Bedir, Z., Sögütcü, E.K., Gölbaşı, Ö. & Rehman, N.U., 2025. Lie Ideals and Homoderivations in Semiprime Rings. Mathematics. Vol. 13, No. 4, 548.
[11] Melaibari, A., Muthana, N. & Al-Kenani, A., 2016. Centrally-extended homoderivations on rings. Gulf Journal of Mathematics. Vol. 4, No. 2
[12] Posner, E.C., 1957. Derivations in prime rings. Proceedings of the American Mathematical Society. Vol. 8, No. 6, 1093–1100.
[13] Rehman, N., Sogutcu, E.K. & Alnoghashi, H.M., 2023. On generalized homoderivations of prime rings. Matematychni Studii. Vol. 60, No. 1, 12–27.
[14] Wang, X.-K., 1994. Derivations in Prime Near-Rings. Proceedings of the American Mathematical Society. Vol. 121, No. 2, 361.
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