Superstability, noetherian rings and pure-semisimple rings

Abstract: We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings. We characterize noetherian rings via superstability of the class of left modules with embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left noetherian. - The class of left $R$-modules with embeddings is superstable. - For every $\lambda \geq |R| + \aleph_0$, there is $\chi \geq \lambda$ such that the class of left $R$-modules with embeddings has uniqueness of limit models of cardinality $\chi$. - Every limit model in the class of left $R$-modules with embeddings is $\Sigma$-injective. We characterize left pure-semisimple rings via superstability of the class of left modules with pure embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left pure-semisimple. - The class of left $R$-modules with pure embeddings is superstable. - There exists $\lambda \geq (|R| + \aleph_0)^+$ such that the class of left $R$-modules with pure embeddings has uniqueness of limit models of cardinality $\lambda$. - Every limit model in the class of left $R$-modules with pure embeddings is $\Sigma$-pure-injective. We think that both equivalences provide evidence that that the notion of superstability could shed light in the understanding of algebraic concepts. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.