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http://dx.doi.org/10.4134/BKMS.b190429

A NOTE ON GENERALIZED DERIVATIONS AS A JORDAN HOMOMORPHISMS  

Chandrasekhar, Arusha (Department of Mathematics Indian Institute of Science Education and Research)
Tiwari, Shailesh Kumar (Department of Mathematics Indian Institute of Science Education and Research)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.3, 2020 , pp. 709-737 More about this Journal
Abstract
Let R be a prime ring of characteristic different from 2. Suppose that F, G, H and T are generalized derivations of R. Let U be the Utumi quotient ring of R and C be the center of U, called the extended centroid of R and let f(x1, …, xn) be a non central multilinear polynomial over C. If F(f(r1, …, rn))G(f(r1, …, rn)) - f(r1, …, rn)T(f(r1, …, rn)) = H(f(r1, …, rn)2) for all r1, …, rn ∈ R, then we describe all possible forms of F, G, H and T.
Keywords
Generalized derivations; Jordan homomorphism; multilinear polynomials; Utumi quotient ring; extended centroid;
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1 V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40 (2012), no. 6, 1918-1932. https://doi.org/10.1080/00927872.2011.553859   DOI
2 N. J. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955), 19-22.
3 T. S. Erickson, W. S. Martindale, 3rd, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49-63. http://projecteuclid.org/euclid.pjm/1102868622   DOI
4 C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar. 14 (1963), 369-371. https://doi.org/10.1007/BF01895723   DOI
5 I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331-341. https://doi.org/10.2307/1992920   DOI
6 B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147-1166. https://doi.org/10.1080/00927879808826190   DOI
7 N. Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, RI, 1964.
8 V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika 17 (1978), no. 2, 220-238, 242-243.
9 T.-K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 27-38.
10 T.-K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057-4073. https://doi.org/10.1080/00927879908826682   DOI
11 T.-K. Lee and W.-K. Shiue, Derivations cocentralizing polynomials, Taiwanese J. Math. 2 (1998), no. 4, 457-467. https://doi.org/10.11650/twjm/1500407017   DOI
12 T.-K. Lee and W.-K. Shiue, Identities with generalized derivations, Comm. Algebra 29 (2001), no. 10, 4437-4450. https://doi.org/10.1081/AGB-100106767   DOI
13 M. F. Smiley, Jordan homomorphisms onto prime rings, Trans. Amer. Math. Soc. 84 (1957), 426-429. https://doi.org/10.2307/1992823   DOI
14 U. Leron, Nil and power-central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103. https://doi.org/10.2307/1997300   DOI
15 W. S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584. https://doi.org/10.1016/0021-8693(69)90029-5   DOI
16 E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. https://doi.org/10.2307/2032686   DOI
17 S. K. Tiwari, Generalized derivations with multilinear polynomials in prime rings, Comm. Algebra 46 (2018), no. 12, 5356-5372. https://doi.org/10.1080/00927872.2018.1468899   DOI
18 S. K. Tiwari, R. K. Sharma, and B. Dhara, Multiplicative (generalized)-derivation in semiprime rings, Beitr. Algebra Geom. 58 (2017), no. 1, 211-225. https://doi.org/10.1007/s13366-015-0279-x   DOI
19 N. Argac and V. De Filippis, Actions of generalized derivations on multilinear polynomials in prime rings, Algebra Colloq. 18 (2011), Special Issue no. 1, 955-964. https://doi.org/10.1142/S1005386711000836   DOI
20 S. K. Tiwari, R. K. Sharma, and B. Dhara, Identities related to generalized derivation on ideal in prime rings, Beitr. Algebra Geom. 57 (2016), no. 4, 809-821. https://doi.org/10.1007/s13366-015-0262-6   DOI
21 M. Bresar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991), no. 1, 89-93. https://doi.org/10.1017/S0017089500008077   DOI
22 A. Asma, N. Rehman, and A. Shakir, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar. 101 (2003), no. 1-2, 79-82. https://doi.org/10.1023/B:AMHU.0000003893.61349.98   DOI
23 K. I. Beidar, W. S.Martindale, III, and A. V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, 196, Marcel Dekker, Inc., New York, 1996.
24 H. E. Bell and L.-C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (1989), no. 3-4, 339-346. https://doi.org/10.1007/BF01953371   DOI
25 M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), no. 2, 385-394. https://doi.org/10.1006/jabr.1993.1080   DOI
26 L. Carini, V. De Filippis, and G. Scudo, Identities with product of generalized derivations of prime rings, Algebra Colloq. 20 (2013), no. 4, 711-720. https://doi.org/10.1142/S1005386713000680   DOI
27 C.-L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723-728. https://doi.org/10.2307/2046841   DOI
28 B. Dhara, Generalized derivations acting on multilinear polynomials in prime rings, Czechoslovak Math. J. 68(143) (2018), no. 1, 95-119. https://doi.org/10.21136/CMJ.2017.0352-16   DOI
29 V. De Filippis and G. Scudo, Generalized derivations which extend the concept of Jordan homomorphism, Publ. Math. Debrecen 86 (2015), no. 1-2, 187-212. https://doi.org/10.5486/PMD.2015.7070   DOI
30 B. Dhara, S. Sahebi, and V. Rahmani, Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals and right ideals, Math. Slovaca 65 (2015), no. 5, 963-974. https://doi.org/10.1515/ms-2015-0065   DOI