{"paper":{"title":"On the existence of unimodular elements and cancellation of projective modules over noetherian and non-noetherian rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Anjan Gupta","submitted_at":"2014-11-03T06:02:21Z","abstract_excerpt":"Let $R$ be a commutative ring of dimension $d$, $S = R[X]$ or $R[X, 1/X]$ and $P$ a finitely generated projective $S$ module of rank $r$. Then $P$ is cancellative if $P$ has a unimodular element and $r \\geq d + 1$. Moreover if $r \\geq \\dim (S)$ then $P$ has a unimodular element and therefore $P$ is cancellative. As an application we have proved that if $R$ is a ring of dimension $d$ of finite type over a Pr\\\"{u}fer domain and $P$ is a projective $R[X]$ or $R[X, 1/X]$ module of rank at least $d + 1$, then $P$ has a unimodular element and is cancellative."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0369","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"}