On the Structure of Some Types of Higher Derivations
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Authors: A. HOSSEINI AND N. UR REHMAN
DOI: 10.46793/KgJMat2401.123H
Abstract:
In this paper we introduce the concepts of higher {Lgn,Rhn}-derivation, higher {gn,hn}-derivation and Jordan higher {gn,hn}-derivation. Then we give a characterization of higher {Lgn,Rhn}-derivations and higher {gn,hn}-derivations in terms of {Lg,Rh}-derivations and {g,h}-derivations, respectively. Using this result, we prove that every Jordan higher {gn,hn}-derivation on a semiprime algebra is a higher {gn,hn}-derivation. In addition, we show that every Jordan higher {gn,hn}-derivation of the tensor product of a semiprime algebra and a commutative algebra is a higher {gn,hn}-derivation. Moreover, we show that there is a one to one correspondence between the set of all higher {Lgn,Rhn}-derivations and the set of all sequences of {LGn,RHn}-derivations. Also, it is presented that if ???? is a unital algebra and {fn} is a generalized higher derivation associated with a sequence {dn} of linear mappings, then {dn} is a higher derivation. Some other related results are also discussed.
Keywords:
Higher {Lgn,Rhn}-derivation, higher {gn,hn}-derivation, Jordan higher {gn,hn}-derivation, generalized higher derivation.
References:
[1] M. Brešar, Jordan {g,h}-derivations on tensor products of algebras, Linear Multilinear Algebra 64(11) (2016), 2199–2207. https://doi.org/10.1080/03081087.2016.1145184
[2] M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 140(4) (1988), 1003–1006. https://doi.org/10.2307/2047580
[3] M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37 (1988), 321–322.
[4] J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 1104–1110. https://doi.org/10.2307/2040004
[5] A. Fošner, Generalized higher derivations on algebras, Ukrainian Math. J. 69 (2018), 1659–1667. https://doi.org/10.1007/s11253-018-1461-8
[6] A. Hosseini and A. Fošner, Identities related to (σ,τ)-Lie derivations and (σ,τ)-derivations, Boll. Unione Mat. Ital. 11 (2018), 395–401. https://doi.org/10.1007/s40574-017-0141-1
[7] A. Hosseini, Characterization of two-sided generalized derivations, Acta Sci. Math. (Szeged) 86 (2020), 577–600. https://doi.org/10.14232/actasm-020-295-8
[8] A. Hosseini, A new proof of Singer-Wermer theorem with some results on {g,h}-derivations, International Journal of Nonlinear Analysis and Applications 11(1) (2020), 453–471. https://doi.org/10.22075/IJNAA.2019.17189.1915
[9] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.
[10] Y. Li and D. Benkovič. Jordan generalized derivations on triangular algebras, Linear Multilinear Algebra 59 (2011), 841–849. https://doi.org/10.1080/03081087.2010.507600
[11] M. Mirzavaziri, Characterization of higher derivations on algebras, Comm. Algebra 38(3) (2010), 75–86. https://doi.org/10.1080/00927870902828751
[12] M. Mirzavaziri and E. O. Tehrani, Generalized higher derivations are sequences of generalized derivations, Journal of Advanced Research in Pure Mathematics 3(1) (2011), 981–987. https://doi.org/10.5373/JARPM.445.052610
[13] J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese J. Math. 11 (2007), 367–370. https://doi.org/10.11650/twjm/1500404694
[14] F. Wei and Z. Xiao, Generalized Jordan derivations on semiprime rings and Its applications in range inclusion problems, Mediterr. J. Math. 8 (2011), 271–291. https://doi.org/10.1007/s00009-010-0081-9
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