Geodesic
On symmetric generalized ($\theta,\eta$)-biderivations of prime rings
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 73-91.

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In this paper, we characterize the actions of symmetric generalized (θ,η)-biderivations and generalized left (θ,η)-biderivations on Lie ideals and ideals of a prime ring 𝒜. It is shown that ℒ (nonzero square-closed Lie ideal of 𝒜) ⊆𝒵(𝒜), whenever traces of these derivations satisfy any of the following conditions:(i) ([l_1,l_2])^Δ=0,(ii) (l_1l_2)^Δ∈𝒵(𝒜),(iii) ([l_1,l_2])^Δ=(l_1)^θ∘(l_2)^Δ,(iv) (l_1)^Δ(l_2)^Δ+(l_1)^η(l_2)^θ∈𝒵(𝒜),(v) a_1((l_1)^Δ(l_2)^Δ+(l_1l_2)^θ)=0,(vi) (l_1)^Δ(l_2)^θ+(l_1)^θ(l_2)^Δ=0,(vii) ([l_1,l_2])^Δ+[(l_1)^Δ,l_2]∈𝒵(𝒜),(viii) (l_1l_2)^Δ±(l_1)^θ(l_2)^Δ+ (l_1l_2)^θ∈𝒵(𝒜), ∀ l_1,l_2∈ℒ,where 0≠ a_1∈𝒜 is a fixed element,Δ is a trace of these biadditive mappings andθ, η are automorphisms of 𝒜.
Keywords: Lie ideals, prime rings, generalized $(\theta,\eta)$-biderivations, generalized left $(\theta,\eta)$-biderivations 
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Dadhwal, Madhu; Devi, Geeta. On symmetric generalized ($\theta,\eta$)-biderivations of prime rings. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 73-91. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a4/

[1] A. Ali, D. Kumar and P. Miyan, On generalized derivations and commutativity of prime and semiprime rings, Hacet. J. Math. Stat. 40(3) (2011) 367–374.

[2] F. Ali and M.A. Chaudhry, On generalized $(\alpha,\beta )$-derivations of semiprime rings, Turk. J. Math. 35 (2011) 399–404. https://doi.org/10.3906/mat-0906-60

[3] M. Ashraf, A. Ali and S. Ali, On Lie ideals and generalized $(\theta,\phi)$-derivations in prime rings, Comm. Algebra 32(8) (2004) 2977–2985. https://doi.org/10.1081/AGB-120039276

[4] M. Ashraf, N. Rehman, S. Ali and M.R. Mozumder, On generalized $(\sigma,\tau)$ -biderivations in rings, Asian European J. Math. 1 (2010) 1–14.

[5] D. Beniss, B. Fahid and A. Mamouni, On Jordan ideals in prime rings with generalized derivations, Comm. Korean Math. Soc. 32(3) (2017) 495–502. https://doi.org/10.4134/CKMS.c160146

[6] J. Bergen, I.N. Herstein and J.W. Kerr, Lie ideals and derivations of prime rings, J. Algebra 71 (1981) 259–267. https://doi.org/10.1016/0021-8693(81)90120-4

[7] M. Brešar, On generalized biderivations and related maps, J. Algebra 172 (1995) 764–786. https://doi.org/10.1006/jabr.1995.1069

[8] M. Brešar, Semiderivations of prime rings, Proc. Am. Math. Soc. 108(4) (1990) 859–860.

[9] M.N. Daif and H.E. Bell, Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci. 15 (1992) 205–206.

[10] B. Dhara and K.G. Pradhan, A note on multiplicative (generalized)-derivations with annihilator conditions, Georgian Math. J. 23(2) (2016) 191–198. https://doi.org/10.1515/gmj-2016-0020

[11] Ö. Gölbaşi and E. Koç, Generalized derivation on Lie ideals in prime rings, Turk. J. Math. 35 (2011) 23–28. https://doi.org/10.3906/mat-0807-27

[12] Ö. Gölbaşi and E. Koç, Notes on commutativity of prime rings with generalized derivation, Comm. Fac. Sci. Univ. Ank. Series A1. 58(2) (2009) 39–46.

[13] B. Hvala, Generalized derivations in rings, Comm. Algebra 26(4) (1998) 1147–1166. https://doi.org/10.1080/00927879808826190

[14] N. Jacobson, Structure of rings, Amer. Math. Soc. Coll. Pub. 37 (Amer. Math. Soc. Providence R.I., 1956).

[15] H. Marubayashi, M. Ashraf, N. Rehman and S. Ali, On generalized $(\alpha, \beta)$-derivations in prime rings, Algebra Colloq. 17 (2010) 865–874. https://doi.org/10.1142/S1005386710000805

[16] N.M. Muthana, Left centralizer traces, generalized biderivations, left bi-multipliers and Jordan biderivations, Aligarh Bull. Math. 26(2) (2007) 33–45.

[17] M.A. Quadri, M.S. Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34(9) (2003) 1393–1396.

[18] C.J. Reddy, S.V. Kumar and K.M. Reddy, Lie ideals with symmetric left biderivation in prime rings, Italian J. Pure Appl. Math. 41 (2019) 158–166.

[19] N. Rehman and A.Z. Ansari, Generalized left derivations acting as homomorphism and anti-homomorphism on Lie ideals of rings, J. Egypt. Math. Soc. 22 (2014) 327–329. https://doi.org/10.1016/j.joems.2013.12.015

[20] N. Rehman, On commutativity of rings with generalized derivation, Math. J. Okayama Univ. 44 (2002) 43–49.

[21] N. Rehman and S. Huang, On Lie ideals and symmetric Generalized $(\alpha,\beta)$- biderivation in Prime ring, Miskolc Math. Notes 20 (2019) 1175–1183. https://doi.org/10.18514/MMN.2019.2450

[22] G.S. Sandhu, S. Ali, A. Boua and D. Kumar, On generalized $(\alpha,\beta)$-derivations and Lie ideals of prime rings, Rend. Circ. Mat. Pal. Series 2 (2021) 1401–1411. https://doi.org/10.1007/s12215-021-00685-9