On limit models and parametrized Noetherian rings

Abstract

We study limit models in the abstract elementary class of modules with embeddings as algebraic objects. We characterize parametrized noetherian rings using the degree of injectivity of certain limit models.

We show that the number of limit models and how close a ring is from being noetherian are inversely proportional.

Theorem 0.1

Let $n\ge 0$ The following are equivalent.

• (1)
  R is left $(<{\aleph }_n)$-noetherian but not left $(<{\aleph }_{n-1})$-noetherian.
• (2)
  The abstract elementary class of modules with embeddings has exactly $n+1$ non-isomorphic λ-limit models for every $\lambda \ge {\left(\mathrm{card}\right(R)+{\aleph }_0)}^{+}$ such that the class is stable in λ.

We further show that there are rings such that the abstract elementary class of modules with embeddings has exactly κ non-isomorphic λ-limit models for every infinite cardinal κ.