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.



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.

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