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Jordan, D.A. and Jordan, C. R.
(1978).
DOI: https://doi.org/10.1112/jlms/s2-18.1.39
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 [2] 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 [2] 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.