{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:T4N2BMFILW3WCWYOCCHJORWUCG","short_pith_number":"pith:T4N2BMFI","schema_version":"1.0","canonical_sha256":"9f1ba0b0a85db7615b0e108e9746d411b2cecef016573754f1c5b8220e0b7b6d","source":{"kind":"arxiv","id":"1211.6872","version":2},"attestation_state":"computed","paper":{"title":"Similarity and commutators of matrices over principal ideal rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexander Stasinski","submitted_at":"2012-11-29T10:47:16Z","abstract_excerpt":"We prove that if R is a principal ideal ring and A\\in\\M_n(R) is a matrix with trace zero, then A is a commutator, that is, A=XY-YX for some X,Y\\in\\M_n(R). This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over Z due to Laffey and Reams, and as a by-product we obtain new simplified proofs of these results. We also establish a normal form for similarity classes of matrices over PIDs, generalising a result of Laffey and Reams. This normal form is a main ingredient in the proof of the result on commutators."},"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":"1211.6872","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-11-29T10:47:16Z","cross_cats_sorted":[],"title_canon_sha256":"a790a6706c23dc544bc24f9e3b9f41b7e5505690023a980f5378ce22b4b7c937","abstract_canon_sha256":"0708c8a00274ba3a5fbdc06e45854e3c32eb1815151e0c295a1735816e0001c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:44.542263Z","signature_b64":"UsgR7XqrlvfvcBORei0x7UF3qETEktiCXkWZEg4pW7SClEW8ZZzcz6V1ylkUFMfniC4LyHcGmFAQ93fGEITtBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f1ba0b0a85db7615b0e108e9746d411b2cecef016573754f1c5b8220e0b7b6d","last_reissued_at":"2026-05-18T03:32:44.541484Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:44.541484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Similarity and commutators of matrices over principal ideal rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexander Stasinski","submitted_at":"2012-11-29T10:47:16Z","abstract_excerpt":"We prove that if R is a principal ideal ring and A\\in\\M_n(R) is a matrix with trace zero, then A is a commutator, that is, A=XY-YX for some X,Y\\in\\M_n(R). This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over Z due to Laffey and Reams, and as a by-product we obtain new simplified proofs of these results. We also establish a normal form for similarity classes of matrices over PIDs, generalising a result of Laffey and Reams. This normal form is a main ingredient in the proof of the result on commutators."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6872","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1211.6872","created_at":"2026-05-18T03:32:44.541621+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.6872v2","created_at":"2026-05-18T03:32:44.541621+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6872","created_at":"2026-05-18T03:32:44.541621+00:00"},{"alias_kind":"pith_short_12","alias_value":"T4N2BMFILW3W","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"T4N2BMFILW3WCWYO","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"T4N2BMFI","created_at":"2026-05-18T12:27:23.164592+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG","json":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG.json","graph_json":"https://pith.science/api/pith-number/T4N2BMFILW3WCWYOCCHJORWUCG/graph.json","events_json":"https://pith.science/api/pith-number/T4N2BMFILW3WCWYOCCHJORWUCG/events.json","paper":"https://pith.science/paper/T4N2BMFI"},"agent_actions":{"view_html":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG","download_json":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG.json","view_paper":"https://pith.science/paper/T4N2BMFI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.6872&json=true","fetch_graph":"https://pith.science/api/pith-number/T4N2BMFILW3WCWYOCCHJORWUCG/graph.json","fetch_events":"https://pith.science/api/pith-number/T4N2BMFILW3WCWYOCCHJORWUCG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG/action/storage_attestation","attest_author":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG/action/author_attestation","sign_citation":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG/action/citation_signature","submit_replication":"https://pith.science/pith/T4N2BMFILW3WCWYOCCHJORWUCG/action/replication_record"}},"created_at":"2026-05-18T03:32:44.541621+00:00","updated_at":"2026-05-18T03:32:44.541621+00:00"}