Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T02:42:29.718Z Has data issue: false hasContentIssue false

On certain subrings of prime rings with derivations

Published online by Cambridge University Press:  09 April 2009

M. Brešar
Affiliation:
University of MariborPF, Koroška160, 62000 Maribor Slovenia
J. Vukman
Affiliation:
University of MariborPF, Koroška160, 62000 Maribor Slovenia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let D be a nonzero derivation of a noncommutative prime ring R, and let U be the subring of R generated by all [D(x), x], x ∞ R. A classical theorem of Posner asserts that U is not contained in the center of R. Under the additional assumption that the characteristic of R is not 2, we prove a more general result stating that U contains a nonzero left ideal of R as well as a nonzero right ideal of R.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Bell, H. E. and Martindale, W. S., ‘Semiderivations and commutativity in prime rings’, Canad. Math. Bull. 31 (1988), 500508.CrossRefGoogle Scholar
[2]Bergen, J., Herstein, I. N., and Kerr, J. W., ‘Lie ideals and derivations in prime rings’, Algebra 71(1981), 259267.CrossRefGoogle Scholar
[3]Brešar, M. and Vukman, J., ‘Jordan derivations on prime rings’, Bull. Austral. Math. Soc. 37 (1988), 321322.CrossRefGoogle Scholar
[4]Brešar, M. and Vukman, J., ‘Orthogonal derivations and an extension of a theorem of Posner’, Rad. Mat. (1989), 237246.Google Scholar
[5]Brešar, M. and Vukman, J., ‘On left derivations and related mappings’, Proc. Amer. Math. Soc. 110 (1990), 716.Google Scholar
[6]Herstein, I. N., Topics in ring theory (Univ. of Chicago Press, Chicago, 1969).Google Scholar
[7]Herstein, I. N., Rings with involution (Univ. of Chicago Press, Chicago, 1976).Google Scholar
[8]Herstein, I. N., ‘A note on derivations’, Canad. Math. Bull. 21(1978), 369370.CrossRefGoogle Scholar
[9]Jacobson, N., Structure of rings, Colloq. Publ. 37 (Amer. Math. Soc., Providence, RI, 1956).CrossRefGoogle Scholar
[10]Martindale, W. S., ‘Prime rings satisfying a generalized polynomial identity’, J. Algebra 12 (1969), 576584.Google Scholar
[11]Posner, E. C., ‘Derivations in prime rings’, Proc. Amer. Math. Soc. 8(1957), 10931100.Google Scholar
[12]Vukman, J., ‘Commuting and centralizing mappings in prime rings’, Proc. Amer. Math. Soc. 109 (1990), 4752.Google Scholar