Deconstructible classes of modules and stability

TL;DR

This paper proves that various classes of modules with specific embedding properties are stable, using an abstract approach that constructs a stable-like independence relation applicable to many module classes.

Contribution

It introduces a general framework showing stability of deconstructible classes of modules with all embeddings, pure embeddings, and RD-embeddings, extending stability results to broad module classes.

Findings

All free and torsion-free modules over any ring are stable.

Modules with projective or flat dimension ≤ n are stable.

Modules with injective dimension ≤ n over right noetherian rings are stable.

Abstract

We show that every deconstructible class of modules with all embeddings, all pure embedding and all RD-embeddings is stable. The argument is presented in the context of abstract classes of modules without amalgamation and the key idea is to construct a stable-like independence relation. In particular, the following classes of modules with all embeddings, all pure embedding and all RD-embeddings are shown to be stable: all free and torsion-free modules over any ring, and for each $n \geq 0$, the classes of all modules of projective and flat dimension $\leq n$ over any ring, and the class of all modules of injective dimension $\leq n$ over any right noetherian ring.