Comparison of Jordan (sigma,tau)- Derivations and Jordan Triple (sigma,tau)- Derivations in Semiprime Rings

Year 2020, , 264 - 272, 25.06.2020

Emine Koç Sögütcü , Öznur Gölbaşı

https://doi.org/10.37094/adyujsci.563317

Abstract

Let R be a 3!-torsion free semiprime ring, τ, σ two endomorphisms of R, d:R→R an additive mapping and L be a noncentral square-closed Lie ideal of R . An additive mapping d:R→R is said to be a Jordan (σ,τ)-derivation if d(x²)=d(x)σ(x)+τ(x)d(x) holds for all x,y∈R. Also, d is called a Jordan triple (σ,τ)-derivation if d(xyx)=d(x)σ(yx)+τ(x)d(y)σ(x)+τ(xy)d(x), for all x,y∈R. In this paper, we proved the following result: d is a Jordan (σ,τ)-derivation if and only if d is a Jordan triple (σ,τ)-derivation.

Keywords

Semiprime ring, Jordan derivation, Jordan triple derivation, (σ τ)-derivation, Jordan (σ τ)-derivation, Jordan triple (σ τ)-derivation

Supporting Institution

Cübap

Project Number

F563

Yarıasal halkalarda Jordan (σ,τ)- Türevler ve Jordan Üçlü (σ,τ)-Türevlerin Karşılaştırılması

Abstract

R bir 3!-torsion free yarıasal halka, � ve � iki endomorfizm, �: � → � toplamsal dönüşüm ve L merkez tarafından kapsanmayan R halkasının bir kare kapalı Lie ideali olsun. �: � → �toplamsal dönüşümü her �, � ∈ � için �(�²) = �(�)�(�) + �(�)�(�) koşulunu sağlıyorsa d dönüşümüne Jordan (�, �) −türev denir. Ayrıca, �: � → � toplamsal dönüşümü her �, � ∈ � için �(���) = �(�)�(��) + �(�)�(�)�(�) + �(��)�(�) koşulunu sağlıyorsa d dönüşümüne Jordan üçlü (�, �) −türev denir. Bu çalışmada, d bir L üzerinde Jordan (�, �) −türev olması için gerek ve yeter koşul d dönüşümünün L üzerinde Jordan üçlü (�, �) −türev olmasıdır sonucu ispatlanmıştır.

Keywords

Yarıasal halka, Jordan türev, Jordan üçlü türev, (σ, τ) −türev, Jordan (σ, τ) −türev, Jordan üçlü (σ, τ) −türev

Project Number

F563

References

• [1] Herstein, I.N., Jordan derivations of prime rings, Proceedings of the American Mathematical Society, 8, 1104-1110, 1957.
• [2] Cusack, J.M., Jordan derivations on rings, Proceedings of the American Mathematical Society, 53, 321-324, 1975.
• [3] Gupta, V., Jordan derivations on Lie ideals of prime and semiprime rings, East-West Journal of Mathematics, 9 (1), 47-51, 2007.
• [4] Jing, W., Lu, S., Generalized Jordan derivations on prime rings and standard opetaror algebras, Taiwanese Journal of Mathematics, 7, 605-613, 2003.
• [5] Vukman, J., A note on generalized derivations of semiprime rings, Taiwanese Journal of Mathematics, 11, 367-370., 2007.
• [6] Rehman, N., Koç Sögütcü, E., Lie idelas and Jordan Triple (α,β)-derivations in rings, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 69(1), 528-539, 2020. (doi: 10.31801/cfsuasmas.549472)
• [7] Herstein, I.N., Topics in ring theory, The University of Chicago Press, Chicago, London, 1969.
• [8] Bresar, M., Jordan mappings of semiprime rings, Journal of Algebra, 127, 218-228, 1989.
• [9] Fošner, M., Ilišević, D., On Jordan triple derivations and related mappings, Mediterranean. Journal of Mathematics, 5, 415-427, 2008.
• [10] Hongan, M., Rehman, N., Al-Omary, R. M., Lie ideals and Jordan triple derivations in rings, Rendiconti del Seminario Matematico della Università di Padova, 125,147-156, 2011. (doi: 10.4171/RSMUP/125-9).