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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1709.08204","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-09-24T14:43:53Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"6e7f64d17354cef2b33b53fcd2adf5edb55956fb78425fdc27d52bd3eb17a9a4","abstract_canon_sha256":"08ad18fa5c0680ff7447e1df124ede7a8d958d12f2dea2361ab454a2174df4ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:50.138198Z","signature_b64":"wvG00HEsBC8YhlUJ3vZiytgmEuO9Q7TulkYueN99GU5OKd1ql3F907A0YpXARDmuxJhPdQbxt+I1YC0zTnCfCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa163b8a978b19285dcae15165c214be84d91baa126376d633dc329e0119ad86","last_reissued_at":"2026-05-18T00:32:50.137453Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:50.137453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1709.08204","created_at":"2026-05-18T00:32:50.137535+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.08204v2","created_at":"2026-05-18T00:32:50.137535+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.08204","created_at":"2026-05-18T00:32:50.137535+00:00"},{"alias_kind":"pith_short_12","alias_value":"VILDXCUXRMMS","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"VILDXCUXRMMSQXOK","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"VILDXCUX","created_at":"2026-05-18T12:31:49.984773+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2","json":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2.json","graph_json":"https://pith.science/api/pith-number/VILDXCUXRMMSQXOK4FIWLQQUX2/graph.json","events_json":"https://pith.science/api/pith-number/VILDXCUXRMMSQXOK4FIWLQQUX2/events.json","paper":"https://pith.science/paper/VILDXCUX"},"agent_actions":{"view_html":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2","download_json":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2.json","view_paper":"https://pith.science/paper/VILDXCUX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.08204&json=true","fetch_graph":"https://pith.science/api/pith-number/VILDXCUXRMMSQXOK4FIWLQQUX2/graph.json","fetch_events":"https://pith.science/api/pith-number/VILDXCUXRMMSQXOK4FIWLQQUX2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2/action/storage_attestation","attest_author":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2/action/author_attestation","sign_citation":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2/action/citation_signature","submit_replication":"https://pith.science/pith/VILDXCUXRMMSQXOK4FIWLQQUX2/action/replication_record"}},"created_at":"2026-05-18T00:32:50.137535+00:00","updated_at":"2026-05-18T00:32:50.137535+00:00"}