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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."},"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":"1411.0369","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2014-11-03T06:02:21Z","cross_cats_sorted":[],"title_canon_sha256":"4cc9455efb0375e1165b4d912eb669bf74164671d1ad93be81f09e09c6243564","abstract_canon_sha256":"19d83cd132d2d65a063810992fc0101bd56dd0f1fe82454fcebc283fbb7307d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:47.485912Z","signature_b64":"Xlstpq2HK3hjC0UZcOr1qWX1kaSIIlE8TWvzRHKzrjFDkF2eGfKljW0MSGo6Svy6afBqEOtOnn0bqkE63ynHBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7e80b9ec867d40168877178f600c9db0fd37d5c92dc07b1cdb50fad824f9fe0","last_reissued_at":"2026-05-18T01:25:47.485488Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:47.485488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $r$. 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