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Right sided ideals and multilinear polynomials with derivation on prime rings. (English) Zbl 1184.16045

Let \(R\) be a prime ring with the center \(Z(R)\) and \(d\) be a nonzero derivation of \(R\). A well-known result proved by E. C. Posner [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)] states that if \([d(x),x]\in Z(R)\) for all \(x\in R\), then \(R\) must be commutative. In [Pac. J. Math. 134, No. 2, 275-297 (1988; Zbl 0614.16028)], C. Lanski extended Posner’s theorem to a Lie ideal. He proved that if \(L\) is a noncommutative Lie ideal of \(R\) such that \([d(x),x]\in Z(R)\) for all \(x\in L\), then \(\text{char\,}R=2\) and \(R\) satisfies \(S_4(x_1,x_2,x_3,x_4)\), the standard identity. In [Proc. Am. Math. Soc. 124, No. 9, 2625-2629 (1996; Zbl 0859.16031)], P.-H. Lee and T.-K. Lee proved that if \([d(f(x_1,x_2,\dots,x_n)),f(x_1,x_2,\dots,x_n)]\in Z(R)\) for all \(x_1,x_2,\dots,x_n\) in some nonzero ideal of \(R\), then \(f(x_1,x_2,\dots,x_n)\) is central-valued on \(R\), except when \(\text{char\,}R=2\) and \(R\) satisfies \(S_4(x_1,x_2,\dots,x_n)\). Recently, V. De Filippis and O. M. Di Vincenzo [in J. Aust. Math. Soc. 76, No. 3, 357-368 (2004; Zbl 1059.16024)] considered the situation \(\delta([d(f(x_1,x_2,\dots,x_n)),f(x_1,x_2,\dots,x_n)])=0\) for all \(x_1,x_2,\dots,x_n\in R\), where \(d\) and \(\delta\) are two derivations of \(R\).
Inspired by the above works, the paper under review obtains the following result. Let \(R\) be an associative prime ring with characteristic different form \(2\), \(Z(R)\) be its center and \(C\) be its extended centroid. Let \(f(x_1,x_2,\dots,x_n)\) be a nonzero polynomial over \(C\) in \(n\) noncommuting variables, \(d\) be a nonzero derivation of \(R\) and \(\rho\) a nonzero right ideal of \(R\). If \([d^2(f(x_1,x_2,\dots,x_n)),f(x_1,x_2,\dots,x_n)]\in Z(R)\) for all \(x_1,x_2,\dots,x_n\in\rho\), then \(\rho C=eRC\) for some idempotent element \(e\) in the socle of \(RC\) and either \(f(x_1,x_2,\dots,x_n)\) is central-valued in \(eRCe\) or \(eRCe\) satisfies the standard identity \(S_4(x_1,x_2,\dots,x_n)\) unless \(d\) is an inner derivation induced by \(b\in Q\) such that \(b^2=0\) and \(b\rho=0\). – This conclusion is really beautiful and generalizes many related results.
Reviewer: Wei Feng (Beijing)

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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References:

[1] K. I. BEIDAR, Rings with generalized identities, Moscow Univ. Math. Bull., 33 (4) (1978), pp. 53-58. Zbl0407.16002 · Zbl 0407.16002
[2] M. BRESÏAR, One-sided ideals and derivations of prime rings, Proc. Amer. Math. Soc., 122 (4) (1994), pp. 979-983. Zbl0820.16032 MR1205483 · Zbl 0820.16032 · doi:10.2307/2161163
[3] C. M. CHANG, Power central values of derivations on multilinear polynomials, Taiwanese J. Math., 7 (2) (2003), pp. 329-338. Zbl1058.16032 MR1978020 · Zbl 1058.16032
[4] C. L. CHUANG, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (3) (1988), pp. 723-728. Zbl0656.16006 MR947646 · Zbl 0656.16006 · doi:10.2307/2046841
[5] C. L. CHUANG - T. K. LEE, Rings with annihilator conditions on multilinear polynomials, Chinese J. Math., 24 (2) (1996), pp. 177-185. Zbl0855.16029 MR1401645 · Zbl 0855.16029
[6] B. FELZENSZWALB, On a result of Levitzki, Canad. Math. Bull., 21 (1978), pp. 241-242. Zbl0395.16029 MR491793 · Zbl 0395.16029 · doi:10.4153/CMB-1978-040-0
[7] V. DE FILIPPIS - O. M. DI VINCENZO, Posner’s second theorem, multilinear polynomials and vanishing derivations, J. Aust. Math. Soc., 76 (2004), pp. 357-368. Zbl1059.16024 MR2053509 · Zbl 1059.16024 · doi:10.1017/S1446788700009915
[8] V. K. KHARCHENKO, Differential identity of prime rings, Algebra and Logic., 17 (1978), pp. 155-168. Zbl0423.16011 MR541758 · Zbl 0423.16011 · doi:10.1007/BF01670115
[9] C. LANSKI, An Engel condition with derivation, Proc. Amer. Math. Soc., 118 (3) (1993), pp. 731-734. Zbl0821.16037 MR1132851 · Zbl 0821.16037 · doi:10.2307/2160113
[10] C. LANSKI, Differential identities, Lie ideals, and Posner’s theorems, Pacific J. Math., 134 (2) (1988), pp. 275-297. Zbl0614.16028 MR961236 · Zbl 0614.16028 · doi:10.2140/pjm.1988.134.275
[11] P. H. LEE - T. K. LEE, Derivations with engel conditions on multilinear polynomials, Proc. Amer. Math. Soc., 124 (9) (1996), pp. 2625-2629. Zbl0859.16031 MR1327023 · Zbl 0859.16031 · doi:10.1090/S0002-9939-96-03351-5
[12] T. K. LEE, Power reduction property for generalized identities of one sided ideals, Algebra Colloquium, 3 (1996), pp. 19-24. Zbl0845.16017 MR1374157 · Zbl 0845.16017
[13] T. K. LEE, Left annihilators characterized by GPIs, Trans. Amer. Math. Soc., 347 (1995), pp. 3159-3165. Zbl0845.16016 MR1286000 · Zbl 0845.16016 · doi:10.2307/2154780
[14] T. K. LEE, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20 (1) (1992), pp. 27-38. Zbl0769.16017 MR1166215 · Zbl 0769.16017
[15] U. LERON, Nil and power central valued polynomials in rings, Trans. Amer. Math. Soc., 202 (1975), pp. 97-103. Zbl0297.16010 MR354764 · Zbl 0297.16010 · doi:10.2307/1997300
[16] W. S. MARTINDALE III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), pp. 576-584. Zbl0175.03102 MR238897 · Zbl 0175.03102 · doi:10.1016/0021-8693(69)90029-5
[17] E. C. POSNER, Derivation in prime rings, Proc. Amer. Math. Soc., 8 (1957), pp. 1093-1100. Zbl0082.03003 MR95863 · Zbl 0082.03003 · doi:10.2307/2032686
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