  Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let
$\Gamma$ be an additive subgroup of $\Bbbk$. Results of Kaplansky-Santharoubane
and Su classify intermediate series representations of the generalised Witt
algebra $W_\Gamma$ in terms of three families, one parameterised by ${\mathbb
A}^2$ and two by ${\mathbb P}^1$. In this note, we use the first family to
construct a homomorphism $\Phi$ from the enveloping algebra $U(W_\Gamma)$ to a
skew extension of ${\Bbbk}[a,b]$. We show that the image of $\Phi$ is contained
in a (double) idealizer subring of this skew extension and that the
representation theory of idealizers explains the three families. We further
show that the image of $U(W_\Gamma)$ under $\Phi$ is not left or right
noetherian, giving a new proof that $U(W_\Gamma)$ is not noetherian.
  We construct $\Phi$ as an application of a general technique to create ring
homomorphisms from shift-invariant families of modules. Let $G$ be an arbitrary
group and let $A$ be a $G$-graded ring. A graded $A$-module $M$ is an
intermediate series module if $M_g$ is one-dimensional for all $g \in G$. Given
a shift-invariant family of intermediate series $A$-modules parametrised by a
scheme $X$, we construct a homomorphism $\Phi$ from $A$ to a skew-extension of
${\Bbbk}[X]$. The kernel of $\Phi$ consists of those elements which annihilate
all modules in $X$.
