Rings of Quotients of Rings of Derivations

Abstract

The concept of a rational extension of a Lie module is defined as in the associative case [1, pp. 81 and 79]. It then follows from [3, Theorem 2.3] that any Lie module possesses a maximal rational extension (a rational completion), unique up to isomorphism. If now L and K are Lie rings with L⊆ K, we call K a (Lie) ring of quotients of L if K, considered as a Lie module over L, is a rational extension of the Lie module LL. Although we do not know if for every Lie ring L its rational completion can be given a Lie ring structure extending that of L (as is the case for associative rings), this is so, in any case, for abelian Lie rings (Propositions 2 and 4).