High powers in endomorphism rings over Dedekind domains

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 ${\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathbb{A}}(M)$ 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.