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

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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.

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Authors and Affiliations

  1. DePaul University, 60614, Chicago, IL, U.S.A.

    Jeffrey Bergen

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  1. Jeffrey Bergen
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Bergen, J. Lie ideals with regular and nilpotent elements and a result on derivations. Rend. Circ. Mat. Palermo 33, 99–108 (1984). https://doi.org/10.1007/BF02844414

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  • DOI: https://doi.org/10.1007/BF02844414

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