  Let $\mathbb{A}$ be a Dedekind domain and $T$ an endomorphism of a
finitely-generated projective $\mathbb{A}$-module. If $T$ is an $s^{th}$ power
in $\mathrm{End}_{\mathbb{A}}(M)$ for $s$ ranging over an infinite set
$\mathcal{S}$ of positive integers, then (a) $T$ decomposes as a direct sum of
the zero operator and an invertible operator on a summand of $M$ and (b) that
summand is semisimple or of finite order if $\mathcal{S}$ is appropriately
large (what this means depends on the structure of the additive and
multiplicative groups of $\mathbb{A}$). This generalizes a result of M.
Cavachi's to the effect that the only non-singular integer matrix that is an
$s^{th}$ power in $M_n(\mathbb{Z})$ for all $s$ is the identity.
