What does it mean that "the action of $\mathbb{Z}$ on $M$ factors through the quotient $\mathbb{Z}/p^n \mathbb{Z}$"? Let $M$ be a $\mathbb{Z}$-module, i.e. $\mathbb{Z}$ acts on $M$.
What does it mean that "the action of $\mathbb{Z}$ on $M$ factors through the quotient $\mathbb{Z}/p^n \mathbb{Z}$" ?
I want to understand it.
Suppose $\rho: \mathbb{Z} \times M \to M$ is defined by $\rho_z(m)=z\cdot m$ for all $z \in \mathbb{Z}$ and $m \in M$. Then "the action of $\mathbb{Z}$ on $M$ factors through the quotient $\mathbb{Z}/p^n \mathbb{Z}$" implies something like

where $\rho_1(z,m)=(z~\pmod{p^n},m)$ and $\rho_2(z~\pmod
{p^n},m)=z\cdot m$.
But still I don't have the motivation and clear understanding of what does mean by "the action of $\mathbb{Z}$ on $M$ factors through the quotient $\mathbb{Z}/p^n \mathbb{Z}$" ?
Any discussion is appreciated.

The group action can equivalently be described as a group homomorphism
$$\rho':\ \Bbb{Z}\ \longrightarrow\ \operatorname{End}(M).$$
That the action factors over $\Bbb{Z}/p^n\Bbb{Z}$ simply means that this homomorphism factors over $\Bbb{Z}/p^n\Bbb{Z}$.
That is to say $p^n\Bbb{Z}$ is contained in the kernel of $\rho'$, or equivalently  $p^n\cdot m=0$ for all $m\in M$.
