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arXiv · 2610.32544

Finite stabilization, socle extensions, and Faith's conjecture

Shaohui Peng·Guocheng Dai
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Abstract

We give a length-two counterexample to the unrestricted product comparison in Bergman's Question 3.1, positive criteria for quasi-injective common modules, and a semidirect-product version of Question 3.2. For semiprimary right self-injective rings, the countable hull construction yields a cyclic two-layer witness, its Baer class, and an opposite-side extension obstruction. We apply this construction over the square-zero ring of the counterexample. The hull quotient of a countable sum of injectives is the endomorphism space modulo finite-rank maps, and computes the full socle-radical defect of a countable product of the regular module. This gives a second counterexample with free cyclic family factors. The original length-two product embeds its defect in the same quotient through an explicit intersection, and all associated extension classes arise from one specified Baer class. The general Faith conjecture remains unresolved.

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Shaohui Peng, Guocheng Dai. 2026-10-23. Finite stabilization, socle extensions, and Faith's conjecture. https://arxiv.org/abs/2610.32544

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