{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:T4N2BMFILW3WCWYOCCHJORWUCG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0708c8a00274ba3a5fbdc06e45854e3c32eb1815151e0c295a1735816e0001c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-11-29T10:47:16Z","title_canon_sha256":"a790a6706c23dc544bc24f9e3b9f41b7e5505690023a980f5378ce22b4b7c937"},"schema_version":"1.0","source":{"id":"1211.6872","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6872","created_at":"2026-05-18T03:32:44Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6872v2","created_at":"2026-05-18T03:32:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6872","created_at":"2026-05-18T03:32:44Z"},{"alias_kind":"pith_short_12","alias_value":"T4N2BMFILW3W","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"T4N2BMFILW3WCWYO","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"T4N2BMFI","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:4edd1d0d6e58e4594143d06a477807b78358463aaab8e4dd93fa755546d8e4e6","target":"graph","created_at":"2026-05-18T03:32:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"Alexander Stasinski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-11-29T10:47:16Z","title":"Similarity and commutators of matrices over principal ideal rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6872","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:576aeb0d1a44229658cec024565db9f34a17ea0d57cc313c6866fd3e039170de","target":"record","created_at":"2026-05-18T03:32:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0708c8a00274ba3a5fbdc06e45854e3c32eb1815151e0c295a1735816e0001c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-11-29T10:47:16Z","title_canon_sha256":"a790a6706c23dc544bc24f9e3b9f41b7e5505690023a980f5378ce22b4b7c937"},"schema_version":"1.0","source":{"id":"1211.6872","kind":"arxiv","version":2}},"canonical_sha256":"9f1ba0b0a85db7615b0e108e9746d411b2cecef016573754f1c5b8220e0b7b6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9f1ba0b0a85db7615b0e108e9746d411b2cecef016573754f1c5b8220e0b7b6d","first_computed_at":"2026-05-18T03:32:44.541484Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:44.541484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UsgR7XqrlvfvcBORei0x7UF3qETEktiCXkWZEg4pW7SClEW8ZZzcz6V1ylkUFMfniC4LyHcGmFAQ93fGEITtBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:44.542263Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.6872","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:576aeb0d1a44229658cec024565db9f34a17ea0d57cc313c6866fd3e039170de","sha256:4edd1d0d6e58e4594143d06a477807b78358463aaab8e4dd93fa755546d8e4e6"],"state_sha256":"f321b01a668899c959262906950054b90b9ed6bc44b7dcf90ce8d52dd8057f04"}