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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A result on derivations
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by Tsiu-Kwen Lee and Jer-Shyong Lin PDF
Proc. Amer. Math. Soc. 124 (1996), 1687-1691 Request permission

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 Bres̆ar’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
  • 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
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 1687-1691
  • MSC (1991): Primary 16W25
  • DOI: https://doi.org/10.1090/S0002-9939-96-03234-0
  • MathSciNet review: 1307545