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.
Categories: math.AC math.NT math.RA