Mathematical analysis demonstrates stability across deconstructible module classes under varied embeddings, indicating broad applicability of stable independence relations without amalgamation.
We show that every deconstructible class of modules with all embeddings, all pure embeddings 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 embeddings and all RD-embeddings are shown to be stable: all free and torsion-free modules over any ring, and for each n ≥ 0, the classes of all modules of projective and flat dimension ≤ n over any ring, and the class of all modules of injective dimension ≤ n over any right noetherian ring.
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Mazari‐Armida et al. (2026) studied this question.
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