High powers in endomorphism rings over Dedekind domains

TL;DR

This paper investigates conditions under which endomorphisms over Dedekind domains are powers of other endomorphisms, revealing their structural decompositions and properties when they are powers for infinitely many integers.

Contribution

It generalizes Cavachi's result by characterizing endomorphisms over Dedekind domains that are powers for infinitely many integers, including their decomposition and semisimplicity.

Findings

Endomorphisms decompose into zero and invertible parts.

Summands are semisimple or of finite order under certain conditions.

Characterization extends Cavachi's integer matrix result.

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 $\mathrm{End}_{\mathbb{A}}(M)$ 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.