nLab p-local module

Let $G$ be a group and $M$ an abelian group with a $G$-action, i.e. a $\mathbb{Z}G$-module where $\mathbb{Z}G$ is the group ring of $G$. Let $T$ be a set of prime numbers; we say that $M$ is a $T$-local module if for all $x\in G$, the element $(1 + x + x^2 + \cdots + x^{n-1})\in \mathbb{Z}G$ acts invertibly on $M$ for all $n$ not divisible by any primes in $T$. If $T=\{p\}$ we speak of a $p$-local module.

Note that if $G$ acts trivially on $M$, this is equivalent to asking that $M$ is a $T$-local abelian group. In general, it is strictly stronger.

The notion can also be characterized as a module over the localization of the (noncommutative!) ring $\mathbb{Z}G$ at the set of elements $1 + x + x^2 + \cdots + x^{n-1}$.