Let $ \mathfrak{S} $ be a prime ring with automorphisms $ \alpha, \beta $. A bi-additive map $ \mathfrak{D} $ is called an ($ \alpha, \beta $) Jordan bi-derivation if $ \mathfrak{D}(k^2, s) = \mathfrak{D}(k, s)\alpha(k) + \beta(k) \mathfrak{D}(k, s) $. In this paper, we find conditions under which a symmetric ($ \alpha, \beta $) Jordan bi-derivation becomes a symmetric ($ \alpha, \beta $) bi-derivation. We also characterize the symmetric $ (\alpha, \beta) $ Jordan bi-derivations.
Citation: Wasim Ahmed, Amal S. Alali, Muzibur Rahman Mozumder. Characterization of $ (\alpha, \beta) $ Jordan bi-derivations in prime rings[J]. AIMS Mathematics, 2024, 9(6): 14549-14557. doi: 10.3934/math.2024707
Let $ \mathfrak{S} $ be a prime ring with automorphisms $ \alpha, \beta $. A bi-additive map $ \mathfrak{D} $ is called an ($ \alpha, \beta $) Jordan bi-derivation if $ \mathfrak{D}(k^2, s) = \mathfrak{D}(k, s)\alpha(k) + \beta(k) \mathfrak{D}(k, s) $. In this paper, we find conditions under which a symmetric ($ \alpha, \beta $) Jordan bi-derivation becomes a symmetric ($ \alpha, \beta $) bi-derivation. We also characterize the symmetric $ (\alpha, \beta) $ Jordan bi-derivations.
| [1] |
I. N. Herstein, Jordan derivation of prime rings, P. Am. Math. Soc., 8 (1957), 1104–1110. https://doi.org/10.1090/S0002-9939-1957-0095864-2 doi: 10.1090/S0002-9939-1957-0095864-2
|
| [2] |
T. K. Lee, Functional identities and Jordan $\sigma$-derivations, Linear Multilinear A., 62 (2016), 221–234. https://doi.org/10.1080/03081087.2015.1032200 doi: 10.1080/03081087.2015.1032200
|
| [3] | G. Maksa, A remark on symmetric biadditive functions having nonnegative diagonalization, Glas. Mat., 15 (1980), 279–282. |
| [4] |
J. Vukman, Symmetric bi-derivations on prime and semiprime rings, Aequationes Math., 38 (1989), 245–254. https://doi.org/10.1007/BF01840009 doi: 10.1007/BF01840009
|
| [5] |
C. Abdioǧlu, T. K. Lee, A basic functional identity with applications to Jordan $\sigma$-biderivations, Commun. Algebra, 45 (2017), 1741–1756. https://doi.org/10.1080/00927872.2016.1222413 doi: 10.1080/00927872.2016.1222413
|
| [6] | M. Ashraf, N. Rehman, S. Ali, On Lie ideals and Jordan generalized derivations of prime rings, Indian J. Pure Ap. Math., 34 (2003), 291–294. |
| [7] |
M. Brešar, Jordan derivations on semiprime rings, P. Am. Math. Soc., 104 (1988), 1003–1006. https://doi.org/10.1090/S0002-9939-1988-0929422-1 doi: 10.1090/S0002-9939-1988-0929422-1
|
| [8] |
M. Brešar, J. Vukman, Jordan derivation on prime rings, B. Aust. Math. Soc., 37 (1988), 321–322. https://doi.org/10.1017/S0004972700026927 doi: 10.1017/S0004972700026927
|
| [9] |
J. M. Cusack, Jordan derivations on rings, P. Am. Math. Soc., 53 (1975), 321–324. https://doi.org/10.1090/S0002-9939-1975-0399182-5 doi: 10.1090/S0002-9939-1975-0399182-5
|
| [10] |
V. D. Filippis, A. Mamouni, L. Oukhtite, Generalized Jordan semiderivations in prime rings, Can. Math. Bull. 58 (2015), 263–270. https://doi.org/10.4153/CMB-2014-066-9 doi: 10.4153/CMB-2014-066-9
|
| [11] |
W. Jing, S. Lu, Generalized Jordan derivations on prime rings and standard operator algebras, P. Am. Math. Soc., 7 (2003), 605–613. https://doi.org/10.11650/twjm/1500407580 doi: 10.11650/twjm/1500407580
|
| [12] |
T. K. Lee, J. H. Lin, Jordan derivations of prime rings with characteristic two, Linear Algebra Appl., 462 (2014), 1–15. https://doi.org/10.1016/j.laa.2014.08.006 doi: 10.1016/j.laa.2014.08.006
|
Wasim Ahmed, Amal S. Alali, Muzibur Rahman Mozumder. Characterization of $ (\alpha, \beta) $ Jordan bi-derivations in prime rings[J]. AIMS Mathematics, 2024, 9(6): 14549-14557. doi: 10.3934/math.2024707
DownLoad: