Finite stabilization, socle extensions, and Faith's conjecture
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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BibTeX
@misc{peng2026finite,
title = {{Finite stabilization, socle extensions, and Faith's conjecture}},
author = {Shaohui Peng and Guocheng Dai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32544},
eprint = {2610.32544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32544},
}
RIS
TY - UNPB
TI - Finite stabilization, socle extensions, and Faith's conjecture
AU - Shaohui Peng
AU - Guocheng Dai
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32544
UR - https://arxiv.org/abs/2610.32544
AB - 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.
ER -