On superstability in the class of flat modules and perfect rings

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