Among the finitely generated modules over a Noetherian ring
$R$, the semidualizing modules have been singled out due to
their particularly nice duality properties. When $R$ is a
normal domain, we exhibit a natural inclusion of the set of
isomorphism classes of semidualizing $R$-modules into the
divisor class group of $R$. After a description of the basic
properties of this inclusion, it is employed to investigate
the structure of the set of isomorphism classes of
semidualizing $R$-modules. In particular, this set is
described completely for determinantal rings over normal
domains.