{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:JUIF5JEC45Q5TB4TGW2FIK33GA","short_pith_number":"pith:JUIF5JEC","canonical_record":{"source":{"id":"1607.07205","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-25T11:08:09Z","cross_cats_sorted":[],"title_canon_sha256":"fb14f5567b16034b6a559dc4c701422841fb632b2b9a1611b298c1fa80ac6f1d","abstract_canon_sha256":"06d8ebd8e6e316feb027745d0aabb6c4b72820ff15739128b7752c9d13e6a20d"},"schema_version":"1.0"},"canonical_sha256":"4d105ea482e761d9879335b4542b7b3038ca9c10a29d2001d66c7cbc6ef9a24c","source":{"kind":"arxiv","id":"1607.07205","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07205","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07205v2","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07205","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"pith_short_12","alias_value":"JUIF5JEC45Q5","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JUIF5JEC45Q5TB4T","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JUIF5JEC","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:JUIF5JEC45Q5TB4TGW2FIK33GA","target":"record","payload":{"canonical_record":{"source":{"id":"1607.07205","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-25T11:08:09Z","cross_cats_sorted":[],"title_canon_sha256":"fb14f5567b16034b6a559dc4c701422841fb632b2b9a1611b298c1fa80ac6f1d","abstract_canon_sha256":"06d8ebd8e6e316feb027745d0aabb6c4b72820ff15739128b7752c9d13e6a20d"},"schema_version":"1.0"},"canonical_sha256":"4d105ea482e761d9879335b4542b7b3038ca9c10a29d2001d66c7cbc6ef9a24c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:29.371040Z","signature_b64":"p4vL//qN/CdNSq+z1ofdJB7CRYUQ4m6yZHLneTkhNgb/8iRhx+r91Xuf5hTFxpm5z3P6g77hGS+A94kcGMhXBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d105ea482e761d9879335b4542b7b3038ca9c10a29d2001d66c7cbc6ef9a24c","last_reissued_at":"2026-05-18T00:50:29.370437Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:29.370437Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.07205","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:50:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TOsf9vTOV2CsV6vUqnlZY0tRIt5SvU9yHSoK4LtxlZgPEijk4lv3EMgkAIb9OVvjtp6EO5QCcWB1sKMHURNVCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:57:24.386069Z"},"content_sha256":"8b63d88c55f893e62a63f143a456f1e10e99f9768c9b974623ba4661b5ea2b92","schema_version":"1.0","event_id":"sha256:8b63d88c55f893e62a63f143a456f1e10e99f9768c9b974623ba4661b5ea2b92"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:JUIF5JEC45Q5TB4TGW2FIK33GA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Commutators of trace zero 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":"2016-07-25T11:08:09Z","abstract_excerpt":"We prove that for every trace zero matrix $A$ over a principal ideal ring $R$, there exist trace zero matrices $X,Y$ over $R$ such that $XY-YX=A$. Moreover, we show that $X$ can be taken to be regular mod every maximal ideal of $R$. This strengthens our earlier result that $A$ is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is significantly simpler than the earlier one.\n  Shalev has conjectured an analogous statement for group commutators in $\\mathrm{SL}_{n}$ over $p$-adic integers. We prove Shalev's conjecture for $n=2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07205","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:50:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DCokeFtOtqq/dC8YGgpqWngN18VmdQIC4RjD0xkGGdO5uPVWonMsgQ6iMlZrZyCKtlFUsUU1NPe3oGqQP7yrDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:57:24.386703Z"},"content_sha256":"8f00507eb3b997bd1b6542434c0ee079858c6d499e2cdcb0027c60303804f5bf","schema_version":"1.0","event_id":"sha256:8f00507eb3b997bd1b6542434c0ee079858c6d499e2cdcb0027c60303804f5bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JUIF5JEC45Q5TB4TGW2FIK33GA/bundle.json","state_url":"https://pith.science/pith/JUIF5JEC45Q5TB4TGW2FIK33GA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JUIF5JEC45Q5TB4TGW2FIK33GA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T16:57:24Z","links":{"resolver":"https://pith.science/pith/JUIF5JEC45Q5TB4TGW2FIK33GA","bundle":"https://pith.science/pith/JUIF5JEC45Q5TB4TGW2FIK33GA/bundle.json","state":"https://pith.science/pith/JUIF5JEC45Q5TB4TGW2FIK33GA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JUIF5JEC45Q5TB4TGW2FIK33GA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:JUIF5JEC45Q5TB4TGW2FIK33GA","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":"06d8ebd8e6e316feb027745d0aabb6c4b72820ff15739128b7752c9d13e6a20d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-25T11:08:09Z","title_canon_sha256":"fb14f5567b16034b6a559dc4c701422841fb632b2b9a1611b298c1fa80ac6f1d"},"schema_version":"1.0","source":{"id":"1607.07205","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07205","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07205v2","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07205","created_at":"2026-05-18T00:50:29Z"},{"alias_kind":"pith_short_12","alias_value":"JUIF5JEC45Q5","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JUIF5JEC45Q5TB4T","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JUIF5JEC","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:8f00507eb3b997bd1b6542434c0ee079858c6d499e2cdcb0027c60303804f5bf","target":"graph","created_at":"2026-05-18T00:50:29Z","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 for every trace zero matrix $A$ over a principal ideal ring $R$, there exist trace zero matrices $X,Y$ over $R$ such that $XY-YX=A$. Moreover, we show that $X$ can be taken to be regular mod every maximal ideal of $R$. This strengthens our earlier result that $A$ is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is significantly simpler than the earlier one.\n  Shalev has conjectured an analogous statement for group commutators in $\\mathrm{SL}_{n}$ over $p$-adic integers. We prove Shalev's conjecture for $n=2$.","authors_text":"Alexander Stasinski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-25T11:08:09Z","title":"Commutators of trace zero matrices over principal ideal rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07205","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":"sha256:8b63d88c55f893e62a63f143a456f1e10e99f9768c9b974623ba4661b5ea2b92","target":"record","created_at":"2026-05-18T00:50:29Z","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":"06d8ebd8e6e316feb027745d0aabb6c4b72820ff15739128b7752c9d13e6a20d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-25T11:08:09Z","title_canon_sha256":"fb14f5567b16034b6a559dc4c701422841fb632b2b9a1611b298c1fa80ac6f1d"},"schema_version":"1.0","source":{"id":"1607.07205","kind":"arxiv","version":2}},"canonical_sha256":"4d105ea482e761d9879335b4542b7b3038ca9c10a29d2001d66c7cbc6ef9a24c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d105ea482e761d9879335b4542b7b3038ca9c10a29d2001d66c7cbc6ef9a24c","first_computed_at":"2026-05-18T00:50:29.370437Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:29.370437Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p4vL//qN/CdNSq+z1ofdJB7CRYUQ4m6yZHLneTkhNgb/8iRhx+r91Xuf5hTFxpm5z3P6g77hGS+A94kcGMhXBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:29.371040Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.07205","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b63d88c55f893e62a63f143a456f1e10e99f9768c9b974623ba4661b5ea2b92","sha256:8f00507eb3b997bd1b6542434c0ee079858c6d499e2cdcb0027c60303804f5bf"],"state_sha256":"a7267586075e4d5835a69f3fbe374af5c060a77f341b2f99d1a54a03bc879d34"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e9vs7ht+QLAfcfEJJMkcJdtXYF1vHv+WNUw7nEHL8bQV4TNVlPbohwKrtWxaY/4WcSJ9aX38A2sdqOIK/zi/AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:57:24.391036Z","bundle_sha256":"93136cc03b193732e0122a19d8beef8d8ddd13b4b979e2038108ca1c303dfb16"}}