Homoderivations and Their Impact on Lie Ideals in Prime Rings

Year 2023, Volume: 6 Issue: 2, 41 - 48, 30.12.2023

Evrim Güven

https://doi.org/10.38061/idunas.1356057

Cited By: 1

Abstract

Assume we have a prime ring denoted as $R$, with a characteristic distinct from two. The concept of a homoderivation refers to an additive map $Η$ of a ring $R$ that satisfies the property $Η(r_1 r_2 )=Η(r_1 ) r_2+r_1 Η(r_2 )+Η(r_1 )Η(r_2 )$, $\forall r_1,r_2 \in R$. This article aims to obtain results for prime rings, ideals, and Lie ideals by utilizing the concept of homoderivation in conjunction with the established theory of derivations.

Keywords

Prime ring, Homoderivation, Lie ideal, Jordan ideal.

References

• 1. Alharfie E. F., Muthana N. M. (2018). The commutativity of prime rings with homoderivations, Int. J. of Adv. and App. Sci., 5(5), 79-81.
• 2. Awtar R. (1984). Lie ideals and Jordan derivations of prime rings, Proc. Amer. Math. Soc., 90, 1, 9-14.
• 3. Atteya M. J. (2022). Homogeneralized (σ,τ)-Derivations of Associative Rings, Studies on Scientific Developments in Geometry, Algebra, and Applied Mathematics, 52.
• 4. Bergen J., Herstein I. N., Kerr J. (1981). Lie Ideals and derivations of Prime Rings, Journal of Algebra, 71, 259-267.
• 5. Divinsky N. (1965). Rings and Radicals, University of Toronto Press, Toronto.
• 6. Ebrahimi M. M., Pajoohesh H. (2003). Inner derivations and homoderivations on ϱ-Rings, Acta Math. Hungar., 100, 157-165.
• 7. El-Soufi M. M. and Ghareeb A. (2022). Centrally Extended α-Homoderivations on Prime and Semiprime Rings, Journal of Mathematics.
• 8. El Sofy, M. M. (2000). Rings with some kinds of mappings, M.Sc. Thesis, Cairo University, Branch of Fayoum, Egypt.
• 9. Engin A., Aydın, N. (2023). Homoderivations in Prime Rings, Journal of New Theory, 43, 23-24.
• 10. Güven E. (2019). Some Results on Left (σ,τ)-Jordan Ideals and one sided Generalized Derivations, TWMS J. App. and Eng. Math., 9, 1, 22-29.
• 11. Herstein, I.N. (1979). A Note On Derivations II, Canad. Math. Bull., 22 (4), 509-511.
• 12. Lee P. H., Lee T.K., (1981). On Derivations of Prime Rings Chinese Journal of Mathematics, 9, 2, 107-110.
• 13. Mayne, J. H. (1984). Centralizing Mappings of Prime Rings, Canadian Mathematical Bulletin 27 (1), 122--126.
• 14. Mouhssine S. and Boua A. (2021). Homoderivations and Semigroup Ideals in 3-Prime Near-Rings, Algebraic Str. and Their App. 8, No. 2, 177-194.
• 15. Posner E. C. (1957). Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100.
• 16. Rehman N., Mozumder M. R., Abbasi A. (2019). Homoderivations on ideals of prime and semiprime rings, The Aligarh Bull. of Math., 38-1, 77-87.
• 17. Aydın N., Kaya K. (1992). Some Generalizations in Prime Rings with (σ,τ)-Derivation, Doğa-Tr. J. Mathematics, 16, 169-176.
• 18. Bresar M. (1991). On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J., 33, 89-93.