The Lie structure of a commutative ring with derivation

Jordan, D.A. and Jordan, C. R. (1978). The Lie structure of a commutative ring with derivation. Journal of the London Mathematical Society, 2(18) pp. 39–49.

DOI: https://doi.org/10.1112/jlms/s2-18.1.39

Abstract

Let $R$ be a commutative ring with identity and δ be a derivation of $R$ . Then the set, $R$ δ, of all derivations of $R$ of the form $r$ δ : x → $r$ δ(x), $r$ ∈ $R$ , is a Lie subring of the Lie ring $D$ (R) of derivations of $R$ . In [2] the authors studied the structure of $D$ (R) and found that $R$ δ played an analogous role to that played by the Lie ring $I$ (S) of inner derivations of a non-commutative ring S in the study of $D$ (S). Furthermore it was shown in [2] that the properties of $R$ δ closely resemble the known properties of $I$ (S). In particular it was shown that the following results hold in the case where $R$ is 2-torsion-free: (i) if $R$ is prime or if $R$ is δ-prime noetherian then $R$ δ is a prime Lie ring; (ii) if $R$ is δ-simple noetherian then $R$ δ is a simple Lie ring. The purpose of this paper is to continue the study of the structure of the Lie ring $R$ δ and of certain of its Lie subrings.

In this paper we shall not view $R$ δ as a Lie subring of $D$ (R) but rather as a Lie ring whose elements are the elements of $R$ with the product [r, s] = rδ(s)—sδ(r) for all r, s ∈ $R$ δ. The two approaches coincide in the case where the annihilator of δ( $R$ ) is zero. It is perhaps worth noting that $R$ δ is isomorphic to a Lie subring of the Lie ring arising from a certain non-commutative ring, namely the Ore extension $S$ = $R$ [x, δ]. The set of those elements of $S$ of the form rx, r ∈ $R$ , is closed under the Lie product in $S$ (that is, [s, t] = st — ts for all s, t∈S) and forms a Lie ring which is isomorphic to $R$ δ.