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### Cocentralizing generalized derivations on multilinear polynomial on right ideals of prime rings

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Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R, f(x1,…,xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(f(r))f(r) – f(r)H(f(r)) € C, for all r = (r1,…,rn) € In. Then one of the following holds: 1. there exist a; b; p (…) ; 2. R satisfies s4, the standard identity of degree 4, and there exist a; (…) ; 3. R satisfies s4 and there exist a; (…) ; 4. R satisfies s4 and there exist a; (…) ; 5. there exists e2 = e (…) and one of the following holds: (a) (…) is an identity for I; (b) char(R) = 2 and s4(x1, x2, x3, x4)x5 is an identity for I; (c) (…) is an identity for I and there exist a, a’, b, b’ (…) , a derivation of R, such that G(x) = ax + xa’ + d(x), H(x) = bx + xb’ – d(x), for all x (…) R, with (a – b’ – α)I = (0) = (b – a’ – α)I.
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22--36
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Bibliogr. 28 poz.
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autor
• Department of Mathematics Belda College Belda, Paschim Medinipur-721424, India, basu_dhara@yahoo.com
Bibliografia
• [1] A. Argac, L. Carini, V. De Filippis, An Engel condition with generalized derivations on Lie ideals, Taiwanese J. Math. 12(2) (2008), 419–433.
• [2] A. Argac, V. De Filippis, Actions of Generalized Derivations on Multilinear Polynomials in Prime Rings, Algebra Colloquium 18 (spec. 1) 2011, 955–964.
• [3] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385–394.
• [4] C. M. Chang, T. K. Lee, Annihilators of power values of derivations in prime rings, Comm. Algebra 26(7) (1998), 2091–2113.
• [5] C. L. Chuang, The additive subgroup generated by a polynomial, Israel J. Math. 59(1) (1987), 98–106.
• [6] C. L. Chuang, GPI’s having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103(3) (1988), 723–728.
• [7] C. L. Chuang, T. K. Lee, Rings with annihilator conditions on multilinear polynomials, Chinese J. Math. 24(2) (1996), 177–185.
• [8] B. Dhara, V. De Filippis, Notes on generalized derivations on Lie ideals in prime rings, Bull. Korean Math. Soc. 46(3) (2009), 599–605.
• [9] O. M. Di Vincenzo, On the n-th centralizer of a Lie ideal, Boll. Un. Mat. Ital. A (7) 3 (1989), 77–85.
• [10] T. S. Erickson, W. S. Martindale III, J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), 49–63.
• [11] C. Faith, Y. Utumi, On a new proof of Litoff’s theorem, Acta Math. Acad. Sci. Hung. 14 (1963), 369–371.
• [12] I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago, IL, 1969.
• [13] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Pub., 37, Amer. Math. Soc., Providence, RI, 1964.
• [14] V. K. Kharchenko, Differential identity of prime rings, Algebra Logic 17 (1978), 155–168.
• [15] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118(3) (1993), 731–734.
• [16] C. Lanski, S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math. 42(1) (1972), 117–135.
• [17] T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20(1) (1992), 27–38.
• [18] T. K. Lee, Left annihilators characterized by GPIs, Trans. Amer. Math. Soc. 347 (1995), 3159–3165.
• [19] T. K. Lee, Power reduction property for generalized identities of one-sided ideals, Algebra Colloq. 3 (1996), 19–24.
• [20] T. K. Lee, Derivations with Engel conditions on polynomials, Algebra Colloq. 5(1) (1998), 13–24.
• [21] T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27(8) (1999), 4057–4073.
• [22] T. K. Lee, W. K. Shiue, Derivations cocentralizing polynomials, Taiwanese J. Math. 2(4) (1998), 457–467.
• [23] P. H. Lee, T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sinica 23 (1995), 1–5.
• [24] U. Leron, Nil and power central valued polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97–103.
• [25] W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584.
• [26] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.
• [27] L. M. Rowen, Polynomial Identities in Ring Theory, Pure and Applied Mathematics 84, Academic Press, New York, 1980.
• [28] T. L. Wong, Derivations with cocentralizing multilinear polynomials, Taiwanese J. Math. 1 (1997), 31–37.
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Bibliografia
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