{"paper":{"title":"Kaplansky's zero divisor and unit conjectures on elements with supports of size $3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Alireza Abdollahi, Zahra Taheri","submitted_at":"2016-12-03T07:19:27Z","abstract_excerpt":"Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group $G$ and a field $\\mathbb{F}$, the group ring $\\mathbb{F}[G]$ has no zero divisors (has no unit with support of size greater than $1$). In this paper, we study possible zero divisors and units in $\\mathbb{F}[G]$ whose supports have size $3$. For any field $\\mathbb{F}$ and all torsion-free groups $G$, we prove that if $\\alpha \\beta=0$ for some non-zero $\\alpha, \\beta \\in \\mathbb{F}[G]$ such that $|supp(\\alpha)|=3$, then $|supp(\\beta)|\\geq 10$. If $\\mathbb{F}=\\mathbb{F}_2$ is the field with 2 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00934","kind":"arxiv","version":1},"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"}