  Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the
first elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a
prime $\wp$, as $\wp$ varies. In particular, we prove the existence of the
density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r =
2$, we also study the second elementary divisor (the exponent) of the reduction
of $\psi$ modulo $\wp$ and prove that, on average, it has a large norm. Our
work is motivated by the study of J.-P. Serre of an elliptic curve analogue of
Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's
study developed by the first author and M. R. Murty.
