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An Engel condition with derivation
for left ideals


Author: Charles Lanski
Journal: Proc. Amer. Math. Soc. 125 (1997), 339-345
MSC (1991): Primary 16W25; Secondary 16N60, 16U80
DOI: https://doi.org/10.1090/S0002-9939-97-03673-3
MathSciNet review: 1363174
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Abstract: We generalize a number of results in the literature by proving the following theorem: Let $R$ be a semiprime ring, $D$ a nonzero derivation of $R$, $L$ a nonzero left ideal of $R$, and let $[x,y]=xy-yx$. If for some positive integers $t_0,t_1,\dots , t_n$, and all $x\in L$, the identity $[[\dots [[D(x^{t_0}),x^{t_1}],x^{t_2}],\dots ],x^{t_n}]=0$ holds, then either $D(L)=0$ or else the ideal of $R$ generated by $D(L)$ and $D(R)L$ is in the center of $R$. In particular, when $R$ is a prime ring, $R$ is commutative.


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  • 1. H. E. Bell and W. S. Martindale, III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1987), 92-101. MR 88h:16044
  • 2. H. E. Bell and I. Nada, On some center-like subsets of rings, Arch. Math. 48 (1987), 381-387. MR 88h:16045
  • 3. M. Bre[??]sar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385-394. MR 94f:16042
  • 4. M. Bre[??]sar, One-sided ideals and derivations of prime rings, Proc. Amer. Math. Soc. 122 (1994), 979-983. MR 95b:16037
  • 5. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), 723-728. MR 89e:16028
  • 6. C. L. Chuang, $^*$-differential identities of prime rings with involution, Trans. Amer. Math. Soc. 316 (1989), 251-279. MR 90b:16018
  • 7. C. L. Chuang and J. S. Lin, On a conjecture by Herstein, J. Algebra 126 (1989), 119-138. MR 90i:16028
  • 8. C. L. Chuang, Hypercentral derivations, J. Algebra 66 (1994), 34-71. MR 95e:16029
  • 9. Q. Deng and H. E. Bell, On derivations and commutativity in semiprime rings, Comm. Algebra 23 (1995), 3705-3713. CMP 95:17
  • 10. T. S. Erickson, W. S. Martindale, III, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), 49-63. MR 52:3264
  • 11. B. Felzenszwalb, On a result of Levitzki, Canad. Math. Bull. 21 (1978), 241-242. MR 58:10992
  • 12. B. Felzenszwalb, Derivations in prime rings, Proc. Amer. Math. Soc. 84 (1982), 16-20. MR 83b:16030
  • 13. I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, 1969. MR 42:6018
  • 14. I. N. Herstein, Rings with involution, University of Chicago Press, Chicago, 1976. MR 56:406
  • 15. N. Jacobson, Lie algebras, Wiley, New York, 1962; reprint, Dover, New York, 1979. MR 26:1345; MR 80k:17001
  • 16. N. Jacobson, PI-algebras, Lecture Notes in Math., Vol. 441, Springer-Verlag, New York, 1975. MR 51:5654
  • 17. V. K. Kharchenko, Differential identities of semiprime rings, Algebra and Logic 18 (1979), 58-80. MR 81f:16052 (of Russian original)
  • 18. C. Lanski, Differential identities in prime rings with involution, Trans. Amer. Math. Soc. 291 (1985), 765-787. MR 87f:16013
  • 19. C. Lanski, Differential identities, Lie ideals, and Posner's theorems, Pacific J. Math. 134 (1988), 275-297. MR 89j:16051
  • 20. C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 731-734. MR 93i:16050
  • 21. C. Lanski, Derivations with nilpotent values on left ideals, Comm. Algebra 22 (1994), 1305-1320. MR 95h:16048
  • 22. W. S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584. MR 39:257
  • 23. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100. MR 20:2361
  • 24. B. Tilly, Derivations whose iterates are zero or invertible on a left ideal, Canad. Math. Bull. 37 (1994), 124-132. MR 94m:16041
  • 25. J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), 47-52. MR 90h:16010

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Additional Information

Charles Lanski
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
Email: clanski@math.usc.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03673-3
Received by editor(s): August 2, 1995
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society

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