  We obtain a characterization of left perfect rings via superstability of the
class of flat left modules with pure embeddings.
  $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent.
  - $R$ is left perfect.
  - The class of flat left $R$-modules with pure embeddings is superstable.
  - There exists a $\lambda \geq (|R| + \aleph_0)^+$ such that the class of
flat left $R$-modules with pure embeddings has uniqueness of limit models of
cardinality $\lambda$.
  - Every limit model in the class of flat left $R$-modules with pure
embeddings is $\Sigma$-cotorsion.
  A key step in our argument is the study of limit models in the class of flat
modules. We show that limit models with chains of long cofinality are cotorsion
and that limit models are elementarily equivalent.
  We obtain a new characterization via limit models of the rings characterized
in [Rot02]. We show that in these rings the equivalence between left perfect
rings and superstability can be refined. We show that the results for these
rings can be applied to extend [She17, 1.2] to classes of flat modules not
axiomatizable in first-order logic.
