Year: 2026
Type: preprint
Abstract: For every odd prime $p$, we construct a finitely presented group $G_p$ and elements $a,b,c\in\mathbb{F}_p[G_p]$ such that $ab=1$, $ac=0$, and $c\ne0$. Thus $\mathbb{F}_p[G_p]$, and the group algebra over every field of characteristic $p$, fails to be directly finite. The construction combines a single exceptional self-product pair with signed bipartite cancellation. New integer matching data realize the exceptional self-intersection while retaining a strict squared-word decay estimate. We verify the adaptations needed for the earlier bounded-path and planar-localization arguments. A minimal disk picture protects the identity coefficient of $c$, so the proof does not require torsion-freeness.
Source: arXiv (Cornell University)
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222935509