GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS IN RIGHT

Let R be a prime ring, f (x, . . . , x) a multilinear polynomial over Cin n noncommuting indeterminates, I a nonzero right ideal of R, andF : R-> R be a nonzero generalized skew derivation of R.Suppose that F (f (r, . . . , rn))f (r, . . . , r)? C, for all r1, . . . , rn? I.If f (x1, . . . , xn) is not central valued on R, then either char(R) = 2and R satisfies s4or one of the following holds:(i) f (x1, . . . , xn)xn+1is an identity for I;(ii) F (I)I = (0);(iii) [f (x1, . . . , xn), xn+1]xn+2is an identity for I, there existb, c, q? Q with q an invertible element such that F (x) =bx- qxq-1c for all x? R, and q-1cI? I. Moreover, inthis case either (b- c)I = (0) or b - c ? C and f(x1, . . . , xn)2is central valued on R.

PDF

___

[1] C¸ . Demir, N. Arga¸c, Prime rings with generalized derivations on right ideals, Algebra Colloq. 18 (Spec. 1), 987-998, 2011.
[2] K.I. Beidar, W.S. Martindale, III, A.V. Mikhalev, Rings with generalized identities(Monographs and Textbooks in Pure and Applied Mathematics, 196, 1996).
[3] M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (2), 385- 394, 1993.
[4] M. Bresar, One-sided ideals and derivations of prime rings, Proc. Amer. Math. Soc. 122 (4), 979-983, 1994.
[5] L. Carini, V. De Filippis, Identities with generalized derivations on prime rings and Banach algebras, Algebra Colloq., 19 (Spec. 1), 971-986, 2012.
[6] C.M. Chang, Power central values of derivations on multilinear polynomials, Taiwanese J. Math. 7 (2), 329-338, 2003.
[7] C.M. Chang, T.K. Lee, Annihilators of power values of derivations in prime rings, Comm. in Algebra 26 (7), 2091-2113, 1998.
[8] J.C. Chang, On the identitity h(x) = af(x)+g(x)b, Taiwanese J. Math. 7 (1), 103-113, 2003.
[9] J. C. Chang, Generalized skew derivations with annihilating Engel conditions, Taiwanese J. Math. 12 (7), 1641-1650, 2008.
[10] J. C. Chang, Generalized skew derivations with nilpotent values on Lie ideals, Monatsh. Math. 161 (2), 155-160, 2010.
[11] H. W. Cheng and F. Wei, Generalized skew derivations of rings, Adv. Math. (China), 35 (2), 237-243, 2006.
[12] C.L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (3), 723-728, 1988.
[13] C.L. Chuang, Differential identities with automorphisms and antiautomorphisms I, J. Algebra 149 (2), 371-404, 1992.
[14] C.L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra 160 (1), 130-171, 1993.
[15] C.L. Chuang, T.K. Lee, Identities with a single skew derivation, J. Algebra 288 (1), 59-77, 2005.
[16] C.L. Chuang, T.K. Lee, Rings with annihilator conditions on multilinear polynomials, Chinese J. Math. 24 (2), 177-185, 1996.
[17] M.C. Chou, C.K. Liu, An Engel condition with skew derivations, Monatsh. Math. 158 (3), 259-270, 2009.
[18] V. De Filippis, F. Wei, Posner's second theorem for skew derivations on left ideals, Houston J. Math. 38 (2), 373-395, 2012.
[19] C. Faith, Y. Utumi, On a new proof of Litoff 's Theorem, Acta Math. Acad. Sci. Hungar 14, 369-371, 1963.
[20] N. Jacobson, Structure of rings (American Mathematical Society Colloquium Publications, Vol. 37. Revised edition American Mathematical Society, Providence, R.I. 1964).
[21] V.K. Kharchenko, Generalized identities with automorphisms, Algebra and Logic 14 (2), 132-148, 1975.
[22] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (3), 731-734, 1993.
[23] T.K. Lee, Derivations with invertible values on a multilinear polynomial, Proc. Amer. Math. Soc. 119 (4), 1077-1083, 1993.
[24] T.K. Lee, Left annihilators characterized by GPIs, Trans. Amer. Math. Soc. 347 (8), 3159- 3165, 1995.
[25] T.K. Lee, Power reduction property for generalized identities of one-sided ideals, Algebra Colloq. 3 (1), 19-24, 1996.
[26] T.K. Lee, Generalized skew derivations characterized by acting on zero products, Pacific J. Math. 216 (2), 293-301, 2004.
[27] T.K. Lee, W.K. Shiue Derivations cocentralizing polynomials, Taiwanese J. Math. 2 (4), 457-467, 1998.
[28] T.K. Lee, Y. Zhou, An identity with generalized derivations, J. Algebra and Appl. 8 (3), 307-317, 2009.
[29] P.H. Lee, T.L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sinica 23 (1), 1-5, 1995.
[30] U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202, 97-103, 1975.
[31] W.S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12, 576-584, 1969.
[32] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8, 1093-1100, 1957.
[33] T.L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3 (4), 369-378, 1996.