Abstract
Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a nonzero derivation of R and G is a generalized derivation of R. If \(G^2(u)d(u)=0\) for all \(u\in f(R)\), then one of the following holds:
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(i)
there exists \(a\in U\) such that \(G(x)=ax\) for all \(x\in R\) with \(a^2=0\),
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(ii)
there exists \(a\in U\) such that \(G(x)=xa\) for all \(x\in R\) with \(a^2=0\).
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References
Beidar, K.I., Martindale, W.S., Mikhalev, V.: Rings with Generalized Identities. Dekker, New York (1996)
Bresar, M., Chebotar, M.A., Semrl, P.: On derivatons of prime rings. Commun. Algebra. 27, 3129–3135 (1999)
Brešar, M.: On the distance of the composition of two derivations to the generalized derivations. Glasgow Math. J. 33, 89–93 (1991)
Carini, L., De Filippis, V., Scudo, G.: Identities with product of generalized skew derivations on multilinear polynomials. Commun. Algebra. 44(7), 3122–3138 (2016)
Carini, L., De Filippis, V., Scudo, G.: Identities with product of generalized derivations of prime rings. Algebra Colloq. 20, 711–720 (2013)
Chuang, C.L.: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103(3), 723–728 (1988)
De Filippis, V., Di Vincenzo, O.M.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra. 40, 1918–1932 (2012)
Erickson, T.S., Martindale, W.S., Osborn, J.M.: Prime nonassociative algebras. Pacific J. Math. 60, 49–63 (1975)
Faith, C., Utumi, Y.: On a new proof of Litoff’s theorem. Acta Math. Acad. Sci. Hung. 14, 369–371 (1963)
Fosner, M., Vukman, J.: Identities with generalized derivations in prime rings. Mediterr. J. Math. 9, 847–863 (2012)
Giambruno, A., Herstien, I.N.: Deivations with nilpotent values Rend. C. Math Palermo. 30, 199–206 (1981)
Jacobson, N.: Structure of rings, Am. Math. Soc. Colloq. Pub., 37, Am. Math. Soc., Providence, RI (1964)
Kharchenko, V.K.: Differential identity of prime rings. Algebra Log. 17, 155–168 (1978)
Lee, T.K.: Generalized derivations of left faithful rings. Commun. Algebra. 27(8), 4057–4073 (1999)
Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sinica. 20(1), 27–38 (1992)
Leron, U.: Nil and power central polynomials in rings. Trans. Am. Math. Soc. 202, 97–103 (1975)
Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra. 12, 576–584 (1969)
Vukman, J.: On \(\alpha\)-derivations of prime and semiprime rings. Demo. Math. 38, 283–290 (2005)
Vukman, J.: Identities with product of \((\alpha, \beta )\)-derivations on prime rings. Demo. Math. 39, 291–298 (2006)
Acknowledgements
The author is very thankful to Dr. S. K. Tiwari, Assistant Professor, Departments of Mathematics, I.I.T Patna for his valuable guidence, suggestions/comments and also thankful to reviewers for his suggestions.
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Yadav, V.K. Identities involving generalized derivations in prime rings. Rend. Circ. Mat. Palermo, II. Ser 71, 259–270 (2022). https://doi.org/10.1007/s12215-021-00593-y
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DOI: https://doi.org/10.1007/s12215-021-00593-y