On limit models and parametrized noetherian rings

TL;DR

This paper investigates the relationship between limit models in modules and parametrized noetherian rings, establishing a connection between the number of limit models and the rings' proximity to being noetherian.

Contribution

It characterizes parametrized noetherian rings via the degree of injectivity of limit models and links the number of limit models to the rings' noetherian properties.

Findings

Number of limit models inversely related to how close a ring is to being noetherian.

Characterization of rings with exactly n+1 limit models for certain cardinals.

Existence of rings with exactly κ limit models for every infinite cardinal κ.

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. $\textbf{Theorem.}$ Let $n \geq 0$ 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 $\lambda$-limit models for every $\lambda \geq (\operatorname{card}(R) + \aleph_0)^+$ such that the class is stable in $\lambda$. We further show that there are rings such that the abstract elementary class of modules with embeddings has exactly $\kappa$ non-isomorphic $\lambda$-limit models for every infinite cardinal $\kappa$.