Rendiconti del Circolo Matematico di Palermo

, Volume 33, Issue 1, pp 99–108 | Cite as

Lie ideals with regular and nilpotent elements and a result on derivations

  • Jeffrey Bergen
Article
Received:

Abstract

LetU⊄Z be a Lie ideal of a ringR. We examine those ringsR in which everyu∈U is either regular or nilpotent and prove that ifR has no non-zero nil left ideals then eitherR is a domain or an order in the 2×2 matrices over a field. We proceed by first examining ringsR with no non-zero nil left ideals possessing a derivationd≠0 such thatd (x) is nilpotent or invertible, for allx∈R. It is shown that such a ring must either be a division ring or the 2×2 matrices over a division ring. We also prove similar results for semiprime rings where the various indices of nilpotence are assumed to be bounded.

Keywords

Left Ideal Prime Ring Division Ring Polynomial Identity Regular Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer 1984

Authors and Affiliations

  • Jeffrey Bergen
    • 1
  1. 1.DePaul UniversityChicagoU.S.A.

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