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http://dx.doi.org/10.4134/BKMS.b190653

ON JORDAN AND JORDAN HIGHER DERIVABLE MAPS OF RINGS  

Liu, Lei (School of Mathematics and statistics Xidian University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.4, 2020 , pp. 957-972 More about this Journal
Abstract
Let 𝓡 be a 2-torsion free unital ring containing a non-trivial idempotent. An additive map 𝛿 from 𝓡 into itself is called a Jordan derivable map at commutative zero point if 𝛿(AB + BA) = 𝛿(A)B + B𝛿(A) + A𝛿(B) + 𝛿(B)A for all A, B ∈ 𝓡 with AB = BA = 0. In this paper, we prove that, under some mild conditions, each Jordan derivable map at commutative zero point has the form 𝛿(A) = 𝜓(A) + CA for all A ∈ 𝓡, where 𝜓 is an additive Jordan derivation of 𝓡 and C is a central element of 𝓡. Then we generalize the result to the case of Jordan higher derivable maps at commutative zero point. These results are also applied to some operator algebras.
Keywords
Derivations; Jordan derivable maps; Jordan higher derivable maps; commutative zero points;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 D. Benkovic, Jordan derivations and antiderivations on triangular matrices, Linear Algebra Appl. 397 (2005), 235-244. https://doi.org/10.1016/j.laa.2004.10.017   DOI
2 M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1003-1006. https://doi.org/10.2307/2047580   DOI
3 M. Bresar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh, Sect. A 137 (2007), 9-21.   DOI
4 M. Ferrero and C. Haetinger, Higher derivations and a theorem by Herstein, Quaest. Math. 25 (2002), no. 2, 249-257. https://doi.org/10.2989/16073600209486012   DOI
5 M. Ferrero and C. Haetinger, Higher derivations of semiprime rings, Comm. Algebra 30 (2002), no. 5, 2321-2333. https://doi.org/10.1081/AGB-120003471   DOI
6 I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110. https://doi.org/10.2307/2032688   DOI
7 C. R. Miers, Lie homomorphisms of operator algebras, Pacific J. Math. 38 (1971), 717-735. http://projecteuclid.org/euclid.pjm/1102969919   DOI
8 M. Mirzavaziri, Characterization of higher derivations on algebras, Comm. Algebra 38 (2010), no. 3, 981-987. https://doi.org/10.1080/00927870902828751   DOI
9 P. Ribenboim, Higher derivations of rings. I, Rev. Roumaine Math. Pures Appl. 16 (1971), 77-110.
10 P. Ribenboim, Higher derivations of rings. II, Rev. Roumaine Math. Pures Appl. 16 (1971), 245-272.
11 Z. Xiao and F. Wei, Jordan higher derivations on triangular algebras, Linear Algebra Appl. 432 (2010), no. 10, 2615-2622. https://doi.org/10.1016/j.laa.2009.12.006   DOI
12 A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), 209-214. https://doi.org/10.2307/2036730   DOI
13 F. Wei and Z. Xiao, Generalized Jordan triple higher derivations on semiprime rings, Bull. Korean Math. Soc. 46 (2009), no. 3, 553-565. https://doi.org/10.4134/BKMS.2009.46.3.553   DOI
14 F. Wei and Z. Xiao, Higher derivations of triangular algebras and its generalizations, Linear Algebra Appl. 435 (2011), no. 5, 1034-1054. https://doi.org/10.1016/j.laa.2011.02.027   DOI
15 J.-H. Zhang and W.-Y. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl. 419 (2006), no. 1, 251-255. https://doi.org/10.1016/j.laa.2006.04.015   DOI
16 S. Zhao and J. Zhu, Jordan all-derivable points in the algebra of all upper triangular matrices, Linear Algebra Appl. 433 (2010), no. 11-12, 1922-1938. https://doi.org/10.1016/j.laa.2010.07.006   DOI