Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NOTE ON LOCAL COMMUTATORS IN DIVISION RINGS WITH INVOLUTION

  • Bien, Mai Hoang (Faculty of Mathematics and Computer Science University of Science-VNUHCM)
  • Received : 2018.05.17
  • Accepted : 2018.12.05
  • Published : 2019.05.31
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Abstract

In this paper, we consider a conjecture of I. N. Herstein for local commutators of symmetric elements and unitary elements of division rings. For example, we show that if D is a finite dimensional division ring with involution ${\star}$ and if $a{\in}D^*=D{\setminus}\{0\}$ such that local commutators $axa^{-1}x^{-1}$ at a are radical over the center F of D for every $x{\in}D^*$ with $x^{\star}=x$, then either $a{\in}F$ or ${\dim}_F\;D=4$.

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References

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