# 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.

## Abstract

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

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

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