{"paper":{"title":"Zero divisor and unit elements with support of size 4 in group algebras of torsion free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Alireza Abdollahi, Fatemeh Jafari","submitted_at":"2017-09-24T14:43:53Z","abstract_excerpt":"Kaplansky Zero Divisor Conjecture states that if $G $ is a torsion free group and $ \\mathbb{F} $ is a field, then the group ring $\\mathbb{F}[G]$ contains no zero divisor and Kaplansky Unit Conjecture states that if $G $ is a torsion free group and $ \\mathbb{F} $ is a field, then $\\mathbb{F}[G]$ contains no non-trivial units. The support of an element $ \\alpha= \\sum_{x\\in G}\\alpha_xx$ in $\\mathbb{F}[G] $, denoted by $supp(\\alpha)$, is the set $ \\{x \\in G|\\alpha_x\\neq 0\\} $. In this paper we study possible zero divisors and units with supports of size $ 4 $ in $\\mathbb{F}[G]$. We prove that if\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.08204","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}