  A complex vector space $V$ is a prehomogeneous $G$-module if $G$ acts
rationally on $V$ with a Zariski-open orbit. The module is called etale if
$\dim V=\dim G$. We study etale modules for reductive algebraic groups $G$ with
one-dimensional center. For such $G$, even though every etale module is a
regular prehomogeneous module, its irreducible submodules have to be
non-regular. For these non-regular prehomogeneous modules, we determine some
strong constraints on the ranks of their simple factors. This allows us to show
that there do not exist etale modules for $G=\mathrm{GL}_1\times
S\times\cdots\times S$, with $S$ simple.
