RELATIONS OF MULTIPLICATIVE GENERALIZED (α,β)- REVERSE DERIVATION AND α- COMMUTING MAPS

Year 2023, Volume: 6 Issue: 1, 41 - 49, 04.07.2023

Barış Albayrak

https://doi.org/10.38061/idunas.1260138

Abstract

In this paper, properties of ideal I of semiprime ring R with multiplicative generalized (α,β)- reverse derivation with determined not necessarily additive map d is studied. We generalized previous studies for different derivations to multiplicative generalized (α,β)- reverse derivation F. We show that [β(p),d(p)]I=0 for all p∈I or [d(p)],α(p)]I=0 for all p∈I under the given different conditions. Also, we give the relationship between map d and anti-automorphism α of semiprime ring R and automorphism β of semiprime ring R. Under the given different conditions, we examine whether d is α- commuting on ideal I or β- commuting on ideal I and obtain new results.

Keywords

Reverse derivation, semiprime ring, commuting map

References

• Bresar, M. (1991). On the distance of the composition of two derivations to the generalized derivations. Glaskow Math. J., 33,89-93.
• Chang, J. C. (2009). Right generalized (α,β)- derivations having power central values. Taiwanese J. Math., 13(4), 1111-1120.
• Herstein, I. N. (1957). Jordan Derivation of Prime Rings. Proc. Amer. Math. Soc. 8, 1104-1110.
• Dhara, B. and Ali, S. (2013). On Multiplicative (generalized) derivative in prime and semi-prime rings”, Aequat. Math., 86(1-2), 65-79.
• Daif, M.N. (1991) When is a multiplicative derivation additive. Int. J. Math. Sci., 14(3), 615-618.
• Daif, M.N. and Tammam El Sayiad M.S. (2007). Multiplicative generalized Derivation which are additive. East-west J. Math. 9(1), 31-37.
• Tiwari, S.K., Sharma, R.K. and Dhara, B. (2008). Some theorems of commutatively on semiprime ring with mapping. Southeast Asian Bull. Math., 42(2), 279-292
• Alhaidary, Z. S. M. and Majeed, A. H. (2021). Square closed Lie İdeals and Multiplicative (Generalized) (α,β)-reverse derivation of Prime Rings. Journal of Discrete Math. Sci. and Cryptography, 24(7), 2037-2046.
• Malleswari, G. N., Sreenivasulu, S. and Shobhalatha. G. (2021). Semiprime rings with multiplicative (Generalized) – derivations involving left multipliers. Create. Math. Inform, 30(1), 61-68.
• Hongan, M., Rehman, N. and Al-Omary R. M. (2011). Lie ideals and Jordan triple derivations in rings. Rend. Sem. Mat. Univ. Padova, 125.