Lie rings of derivations of associative rings

Jordan, C. R. and Jordan, D. A. (1978). Lie rings of derivations of associative rings. Journal of the London Mathematical Society, 2(17) pp. 33–41.

DOI: https://doi.org/10.1112/jlms/s2-17.1.33

Abstract

Let $R$ be an associative ring with centre $Z$ . The aim of this paper is to study how the ideal structure of the Lie ring of derivations of $R$ , denoted $D(R)$ , is determined by the ideal structure of $R$ . If $R$ is a simple (respectively semisimple) finite-dimensional $Z$ -algebra and δ $(z)$ = 0 for all δ ∈ $D(R)$ , then every derivation of $R$ is inner and $D(R)$ is known to be a simple (respectively semisimple) Lie algebra (see [7, 5]). Here we are interested in extending these results to the case where $R$ is a prime or semi-prime ring.