{"paper":{"title":"On graphs and valuations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Andrei S. Rapinchuk, Ido Efrat, Yoav Segev","submitted_at":"2014-01-31T16:18:57Z","abstract_excerpt":"In the last two decades new techniques emerged to construct valuations on an infinite division ring $D,$ given a normal subgroup $N\\subseteq D$ of finite index. These techniques were based on the commuting graph of $D^{\\times}/N$ in the case where $D$ is non-commutative, and on the Milnor K-graph on $D^{\\times}/N,$ in the case where $D$ is commutative. In this paper we unify these two approaches and consider V-graphs on $D^{\\times}/N$ and how they lead to valuations. We furthermore generalize previous results to situations of finitely many valuations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.8213","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"}