I am reading A. Fröhlich's Formal Groups, and I am working on the proof that if $F$ is a formal group defined over a separably closed field $k$ of characteristic $p$, then the endomorphism ring $E$ of $F$ over $k$ is isomorphic to the maximal order in the central division algebra of invariant $1/h$ and of rank $h^2$ over $\mathbb{Q}_p$, where $h$ is the height of $F$.
One step in the proof is to show that $\mathrm{cent}(E)=\mathbb{Z}_p$. We have shown by this time that $E$ is a complete topological $\mathbb{Z}_p$-module, and is moreover a free $\mathbb{Z}_p$-module of rank $h^2$. The book states: "If $f\in E$, $pf\in\mathrm{cent}(E)$, then $f\in\mathrm{cent}(E)$. Thus $\mathrm{cent}(E)$ is a direct summand of $E$...." How does the second sentence follow from the first? This book is normally very careful to include all the details of an argument, so I'm guessing this follows from a simple fact of ring theory.
If $M$ is a finite rank free $\mathbb Z_p$-module (like $E$) and $N$ is a submodule (like cent$(E)$) such that $M/N$ is torsion-free (which is a rephrasing of the condition $pf \in N$ implies $f \in N$), then $M/N$ is torsion-free and f.g., thus free, and so the surjection $M \to M/N$ splits, showing in particular that $N$ is a direct summand of $M$.
The usual word for a submodule such that the quotient is torsion-free is saturated, and using saturatedness as a technique for producing direct summands (and related ideas) is a standard way of arguing with modules over DVR's.