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Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R."},"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":"1806.04747","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-06-12T20:13:14Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"b0ecf5bce895a68871b2d6a893d0e2b549815ed6fff0f2449e526b5abbed21bd","abstract_canon_sha256":"31d7f08112eee5d54eefdc699d5edcde9345a1793275943340d722ef1a9114c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:20.982285Z","signature_b64":"CzsQbuWgoRn/gpQ4/YSb2lKTX6p1oMw2XjtsYTXIartS69gW9ghFwBzBvO92iytmhjE/UfvdN4HvyEstw0dsDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf67722252fe51aad950959dc7de1420859093b3f70ec67eaf5a33c732eaccab","last_reissued_at":"2026-05-18T00:13:20.981764Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:20.981764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutative Bezout domains of stable range 1.5","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.RA","authors_text":"Victor A. Bovdi, Volodymyr P. Shchedryk","submitted_at":"2018-06-12T20:13:14Z","abstract_excerpt":"A ring R is said to be of stable range 1.5 if for each a, b from R and nonzero c from R satisfying aR + bR + cR = R there exists r from R such that (a + br)R + cR = R. Let R be a commutative domain in which all finitely generated ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. 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