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An Engel condition with derivation. (English) Zbl 0821.16037
Let $$R$$ be an associative prime ring and $$d$$ a nonzero derivation of $$R$$. The author proves the following two generalizations of Posner’s theorem: i) If $$I$$ is a nonzero ideal of $$R$$ and there exists an integer $$k>0$$ such that $$[d(r),\underbrace{r,r,\dots,r}_k]=0$$ for all $$r\in I$$, then $$R$$ is commutative; ii) If $$I$$ is a noncommutative Lie ideal of $$R$$ and there exists an integer $$k>0$$ such that $$[d(r),\underbrace{r,\dots,r}_k]=0$$ for all $$r\in I$$, then $$R\subseteq M_2(F)$$ where $$F$$ is a field of characteristic two.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16U80 Generalizations of commutativity (associative rings and algebras) 16N60 Prime and semiprime associative rings 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16R50 Other kinds of identities (generalized polynomial, rational, involution)
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##### References:
 [1] Chen-Lian Chuang, *-differential identities of prime rings with involution, Trans. Amer. Math. Soc. 316 (1989), no. 1, 251 – 279. · Zbl 0676.16011 [2] Chen-Lian Chuang and Jer-Shyong Lin, On a conjecture by Herstein, J. Algebra 126 (1989), no. 1, 119 – 138. · Zbl 0688.16036 · doi:10.1016/0021-8693(89)90322-0 · doi.org [3] I. N. Herstein, Topics in ring theory, The University of Chicago Press, Chicago, Ill.-London, 1969. · Zbl 0232.16001 [4] Nathan Jacobson, \?\?-algebras, Lecture Notes in Mathematics, Vol. 441, Springer-Verlag, Berlin-New York, 1975. An introduction. · Zbl 0314.15001 [5] V. K. Harčenko, Differential identities of prime rings, Algebra i Logika 17 (1978), no. 2, 220 – 238, 242 – 243 (Russian). [6] Charles Lanski, Differential identities in prime rings with involution, Trans. Amer. Math. Soc. 291 (1985), no. 2, 765 – 787. · Zbl 0581.16008 [7] Charles Lanski, Differential identities, Lie ideals, and Posner’s theorems, Pacific J. Math. 134 (1988), no. 2, 275 – 297. · Zbl 0614.16028 [8] Wallace S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576 – 584. · Zbl 0175.03102 · doi:10.1016/0021-8693(69)90029-5 · doi.org [9] Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. · Zbl 0662.16001 [10] Edward C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093 – 1100. · Zbl 0082.03003 [11] Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. · Zbl 0461.16001 [12] J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), no. 1, 47 – 52. · Zbl 0697.16035
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