Which commutative rings $R$ have the property that there exists a finite non-trivial abelian group $G$ that admits an $R$-module structure?
Any counterexample?
2026-04-18 06:42:19.1776494539
$R$-module structure on finite group
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I assume $R$ has $1$, otherwise the task is trivial.
Suppose $G$ is a nonzero finite $R$-module. Then, if $g\in G$, $g\ne0$, $gR\ne0$ and $gR$ is finite. Thus $R/\operatorname{Ann}_R(g)$ is a finite ring.
So the answer is: if and only if $R$ has an ideal of finite index, which of course is equivalent to “$R$ has a maximal ideal $I$ such that $R/I$ is a finite field”.