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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03673-3
PII: S 0002-9939(97)03673-3
Received by editor(s): August 2, 1995
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society