  Let $G$ be a group which is either virtually soluble or virtually free, and
let $\omega$ be a weight on $G$. We prove that, if $G$ is infinite, then there
is some maximal left ideal of finite codimension in the Beurling algebra
$\ell^1(G, \omega)$ which fails to be (algebraically) finitely generated. This
implies that a conjecture of Dales and Zelazko holds for these Banach algebras.
We then go on to give examples of weighted groups for which this property fails
in a strong way. For instance we describe a Beurling algebra on an infinite
group in which every left ideal of finite codimension is finitely generated,
and which has many such ideals in the sense of being residually finite
dimensional. These examples seem to be hard cases for proving Dales and
Zelazko's conjecture.
