HIGH POWERS IN ENDOMORPHISM RINGS OVER DEDEKIND DOMAINS

Abstract

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 ${\text{End}}_{\mathbb{𝔸}}(M)$ 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 Cavachi’s to the effect that the only nonsingular integer matrix that is an $s$-th power in $M_n(\mathbb{Z})$ for all $s$ is the identity.