  In this paper, we first study some properties peculiar to $C_{0}$--semigroups
of continuous module homomorphisms and give a characterization for such a
$C_{0}$--semigroup to be almost surely bounded. Then, based on these, we
establish the Hille-Yosida generation theorem for almost surely bounded
$C_{0}$--semigroups of continuous module homomorphisms, which generalizes some
known results. Moreover, the counterexample constructed in this paper also
shows that it is necessary to require the almost sure boundedness for such
$C_{0}$--semigroups.
