  Let $A$ be a finite dimensional commutative associative algebra with unit
over an algebraically closed field of characteristic zero. The group $G(A)$ of
invertible elements is open in $A$ and thus $A$ has a structure of a
prehomogeneous $G(A)$-module. We show that every prehomogeneous module of a
commutative linear algebraic group appears this way. In particular, the number
of equivalence classes of prehomogeneous $G$-modules is finite if and only if
the corank of $G$ is at most $5$.

Representation Theory