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

 

A result on derivations


Authors: Tsiu-Kwen Lee and Jer-Shyong Lin
Journal: Proc. Amer. Math. Soc. 124 (1996), 1687-1691
MSC (1991): Primary 16W25
MathSciNet review: 1307545
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Abstract: Let $R$ be a semiprime ring with a derivation $d$ and let $U$ be a Lie ideal of $R$, $a\in R$. Suppose that $ad(u)^n=0$ for all $u\in U$, where $n$ is a fixed positive integer. Then $ad(I)=0$ for $I$ the ideal of $R$ generated by $[U,U]$ and if $R$ is 2-torsion free, then $ad(U)=0$. Furthermore, $R$ is a subdirect sum of semiprime homomorphic images $R_1$ and $R_2$ with derivations $d_1$ and $d_2$, induced canonically by $d$, respectively such that $\overline ad_1(R_1)=0$ and the image of $U$ in $R_2$ is commutative (central if $R$ is 2-torsion free), where $\overline a$ denotes the image of $a$ in $R_1$. Moreover, if $U=R$, then $ad(R)=0$. This gives Bre[??]sar's theorem without the $(n-1)!$-torsion free assumption on $R$.


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Additional Information

Tsiu-Kwen Lee
Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan 10764, Republic of China
Email: tklee@math.ntu.edu.tw

Jer-Shyong Lin
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Email: jslin@math.nthu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03234-0
PII: S 0002-9939(96)03234-0
Keywords: Semiprime rings, derivations, Lie ideals, GPIs, differential identities
Received by editor(s): March 28, 1994
Received by editor(s) in revised form: May 9, 1994, and December 9, 1994
Communicated by: Ken Goodearl
Article copyright: © Copyright 1996 American Mathematical Society