  We study the behaviour of modules $M$ that fit into a short exact sequence
$0\to M\to C\to M\to 0$, where $C$ belongs to a class of modules $\mathcal C$,
the so-called $\mathcal C$-periodic modules. We find a rather general framework
to improve and generalize some well-known results of Benson and Goodearl and
Simson. In the second part we will combine techniques of hereditary cotorsion
pairs and presentation of direct limits, to conclude, among other applications,
that if $M$ is any module and $C$ is cotorsion, then $M$ will be also
cotorsion. This will lead to some meaningful consequences in the category
$\textrm{Ch}(R)$ of unbounded chain complexes and in Gorenstein homological
algebra. For example we show that every acyclic complex of cotorsion modules
has cotorsion cycles, and more generally, every map $F\to C$ where $C$ is a
complex of cotorsion modules and $F$ is an acyclic complex of flat cycles, is
null-homotopic. In other words, every complex of cotorsion modules is
dg-cotorsion.
