Year: 2026
Type: preprint
Abstract: We give a negative answer to the unrestricted product version of Bergman's finite-comparison theorem, even when all factors have finite length. Finite comparison does hold for quasi-injective common modules under finite cogeneration. We also prove a semidirect-product case of Bergman's Question 3.2. For a semiprimary right self-injective ring, failure of the quasi-Frobenius conclusion produces a cyclic module with simple top and countably infinite homogeneous socle inside a fixed countable injective hull. We identify its extension class and an associated opposite-side extension obstruction. Over the square-zero algebra $k\oplus k^{(\mathbb N)}$, we compute the countable hull quotient as endomorphisms modulo finite-rank maps. This quotient also describes the socle-radical quotient of the regular product, yields a second counterexample with free cyclic family factors, and detects splitting of specified simple submodules. The general Faith conjecture remains unresolved.
Source: arXiv (Cornell University)
Authors: Peng, Shaohui, Dai, Guocheng
Institution: Sichuan Normal University
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223680145