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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Smash products and differential identities
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by Chen-Lian Chuang and Yuan-Tsung Tsai PDF
Trans. Amer. Math. Soc. 364 (2012), 4155-4168 Request permission


Let $\mathbf {U}$ be the universal enveloping algebra of a Lie algebra and $R$ a $\mathbf {U}$-module algebra, where $\mathbf {U}$ is considered as a Hopf algebra canonically. We determine the centralizer of $R$ in $R\#\mathbf {U}$ with its associated graded algebra. We then apply this to the Ore extension $R[X;\phi ]$, where $\phi :X\to \mathrm {Der}(R)$. With the help of PBW-bases, the following is proved for a prime ring $R$: Let $Q$ be the symmetric Martindale quotient ring of $R$. For $f_i,g_i\in Q[X;\phi ]$, $\sum _if_irg_i=0$ for all $r\in R$ iff $\sum _if_i\otimes g_i=0$, where $\otimes$ is over the centralizer of $R$ in $Q[X;\phi ]$. Finally, we deduce from this Kharchenko’s theorem on differential identities.
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Additional Information
  • Chen-Lian Chuang
  • Affiliation: Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
  • Email:
  • Yuan-Tsung Tsai
  • Affiliation: Department of Applied Mathematics, Tatung University, Taipei 104, Taiwan
  • Email:
  • Received by editor(s): May 4, 2010
  • Received by editor(s) in revised form: August 30, 2010
  • Published electronically: March 21, 2012

  • Dedicated: To Pjek-Hwee Lee on his retirement
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 4155-4168
  • MSC (2010): Primary 16S40, 16S32, 16W25, 16S36, 16S30
  • DOI:
  • MathSciNet review: 2912449