{"paper":{"title":"On Common Divisors of Fox Derivatives with towards to Zero Divisors of Group Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Viktor Lopatkin","submitted_at":"2018-08-15T14:10:33Z","abstract_excerpt":"Using Composition--Diamond Lemma we construct presentations of groups $G = \\langle x_1,\\ldots,x_n \\, | \\, r_1,\\ldots, r_m \\rangle$ with the following property; for a fixed $1 \\le i \\le n$, and for all $1 \\le j \\le m$, Fox derivatives $\\partial r_j / \\partial x_i$ have common divisor. It follows that if $\\pi_2(K) \\ne 0$, where $K$ is the standard $2$-complex associated with $G$ then the group ring $\\mathbb{Z}[G]$ has nontrivial zero divisors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05087","kind":"arxiv","version":4},"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"}