On limit models and parametrized noetherian rings

TL;DR

This work connects the structure of limit models in the abstract elementary class of modules with embeddings to classical ring-theoretic properties. By developing a framework of parametrized injective modules and left (<κ)-noetherian rings, it derives exact correspondences between noetherian depth and the quantity of limit models, and shows how limit-model injectivity reflects the ring’s algebraic complexity. The results yield precise counts of limit models for both finite and infinite noetherian depths, including the existence of rings with exactly κ limit models for any infinite κ, thus providing new examples of strictly stable AECs. Overall, the paper reveals a deep algebraic substrate for model-theoretic phenomena in module-theoretic AECs and advances understanding of how parametrized injectivity governs limit-model behavior.

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 $λ$-limit models for every $λ\geq (\operatorname{card}(R) + \aleph_0)^+$ 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 $κ$.

Theorems & Definitions (52)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Example 2.6
• Example 2.7: ls
• Remark 2.8
• Definition 2.10