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# Article

MSC: 16N60, 16W25
Full entry | Fulltext not available (moving wall 24 months)
Keywords:
generalized skew derivation; prime ring
Summary:
Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $p(x_1,\ldots ,x_n)$ is a non-central polynomial over $C$ such that $$[F(x),\alpha (y)]=G([x,y])$$ for all $x,y \in \{p(r_1,\ldots ,r_n)\colon r_1,\ldots ,r_n \in R\}$. Then there exists $\lambda \in C$ such that $F(x)=G(x)=\lambda \alpha (x)$ for all $x\in R$.
References:
[1] Arga{ç}, N., Carini, L., Filippis, V. De: An Engel condition with generalized derivations on Lie ideals. Taiwanese J. Math. 12 (2008), 419-433. MR 2402125
[2] Brešar, M., Miers, C. R.: Strong commutativity preserving maps of semiprime rings. Can. Math. Bull. 37 (1994), 457-460. DOI 10.4153/CMB-1994-066-4 | MR 1303671
[3] Chang, J.-C.: On the identity $h(x)=af(x)+g(x)b$. Taiwanese J. Math. 7 (2003), 103-113. MR 1961042
[4] Chuang, C.-L.: Identities with skew derivations. J. Algebra 224 (2000), 292-335. DOI 10.1006/jabr.1999.8052 | MR 1739582
[5] Chuang, C.-L.: Differential identities with automorphisms and antiautomorphisms. II. J. Algebra 160 (1993), 130-171. DOI 10.1006/jabr.1993.1181 | MR 1237081
[6] Chuang, C.-L.: Differential identities with automorphisms and antiautomorphisms. I. J. Algebra 149 (1992), 371-404. DOI 10.1016/0021-8693(92)90023-F | MR 1172436
[7] Chuang, C.-L.: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988), 723-728. DOI 10.1090/S0002-9939-1988-0947646-4 | MR 0947646 | Zbl 0656.16006
[8] Chuang, C.-L.: The additive subgroup generated by a polynomial. Isr. J. Math. 59 (1987), 98-106. DOI 10.1007/BF02779669 | MR 0923664
[9] Chuang, C.-L., Lee, T.-K.: Identities with a single skew derivation. J. Algebra 288 (2005), 59-77. DOI 10.1016/j.jalgebra.2003.12.032 | MR 2138371
[10] Chuang, C.-L., Lee, T.-K.: Rings with annihilator conditions on multilinear polynomials. Chin. J. Math. 24 (1996), 177-185. MR 1401645
[11] Filippis, V. De: A product of two generalized derivations on polynomials in prime rings. Collect. Math. 61 (2010), 303-322. DOI 10.1007/BF03191235 | MR 2732374
[12] Vincenzo, O. M. Di: On the $n$th centralizer of a Lie ideal. Boll. Unione Mat. Ital., VII. Ser. 3-A (1989), 77-85. MR 0990089
[13] Faith, C., Utumi, Y.: On a new proof of Litoff's theorem. Acta Math. Acad. Sci. Hung. 14 (1963), 369-371. DOI 10.1007/BF01895723 | MR 0155858 | Zbl 0147.28602
[14] Herstein, I. N.: Topics in Ring Theory. Chicago Lectures in Mathematics The University of Chicago Press, Chicago (1969). MR 0271135 | Zbl 0232.16001
[15] Jacobson, N.: PI-Algebras: An Introduction. Lecture Notes in Mathematics 441 Springer, Berlin (1975). MR 0369421
[16] Jacobson, N.: Structure of Rings. American Mathematical Society Colloquium Publications 37 AMS, Providence (1956). MR 0081264 | Zbl 0073.02002
[17] Lanski, C., Montgomery, S.: Lie structure of prime rings of characteristic 2. Pac. J. Math. 42 (1972), 117-136. DOI 10.2140/pjm.1972.42.117 | MR 0323839
[18] Lin, J.-S., Liu, C.-K.: Strong commutativity preserving maps on Lie ideals. Linear Algebra Appl. 428 (2008), 1601-1609. MR 2388643 | Zbl 1141.16021
[19] Liu, C.-K.: Strong commutativity preserving generalized derivations on right ideals. Monatsh. Math. 166 (2012), 453-465. DOI 10.1007/s00605-010-0281-1 | MR 2925149
[20] Liu, C.-K., Liau, P.-K.: Strong commutativity preserving generalized derivations on Lie ideals. Linear Multilinear Algebra 59 (2011), 905-915. DOI 10.1080/03081087.2010.535819 | MR 2826060
[21] Ma, J., Xu, X. W., Niu, F. W.: Strong commutativity-preserving generalized derivations on semiprime rings. Acta Math. Sin., Engl. Ser. 24 (2008), 1835-1842. DOI 10.1007/s10114-008-7445-0 | MR 2453063
[22] III, W. S. Martindale: Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576-584. DOI 10.1016/0021-8693(69)90029-5 | MR 0238897
[23] Posner, E. C.: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1958), 1093-1100. DOI 10.1090/S0002-9939-1957-0095863-0 | MR 0095863 | Zbl 0082.03003
[24] Wong, T.-L.: Derivations with power-central values on multilinear polynomials. Algebra Colloq. 3 (1996), 369-378. MR 1422975 | Zbl 0864.16031

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