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Define a left-normed commutator $[a_1, a_2, \\dots , a_n]$ $(a_i \\in A)$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \\dots , a_{n-1}, a_n] = [[a_1, \\dots , a_{n-1}], a_n]$ $(n \\ge 3)$. For $n \\ge 2$, let $T^{(n)} (A)$ be the two-sided ideal in $A$ generated by all commutators $[a_1, a_2, \\dots , a_n]$ ($a_i \\in A )$. Define $T^{(1)} (A) = A$.\n  Let $k, \\ell$ be integers such that $k > 0$, $0 \\le \\ell \\le k$. Let $m_1, \\dots , m_k$ be positive integers such that $\\ell$ of them are odd and "},"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":"1610.03136","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-10-11T00:14:15Z","cross_cats_sorted":[],"title_canon_sha256":"97eb134f2f1bb23063e68fde6ae145ad812f707652f838b9b65f2bf16c2696e0","abstract_canon_sha256":"3b2b208c4b613d41fc2fe89751596d70a9a2ae2c53fe0abebd89c055666b6d9f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:48.343019Z","signature_b64":"MvfV+REjUAkJ6NWEmLZCU9AqGtaGvbuSUPZgibOxqkOvhZ+rk4spto0Iljoze8cgRWXlin4D+0R1mOkp7wbnAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad65b95803b4a6362d81c771d955ab0a3d1af0401265e90045af253846a8c62e","last_reissued_at":"2026-05-18T00:13:48.342426Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:48.342426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Products of several commutators in a Lie nilpotent associative algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexei Krasilnikov, Galina Deryabina","submitted_at":"2016-10-11T00:14:15Z","abstract_excerpt":"Let $F$ be a field of characteristic $\\ne 2,3$ and let $A$ be a unital associative $F$-algebra. Define a left-normed commutator $[a_1, a_2, \\dots , a_n]$ $(a_i \\in A)$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \\dots , a_{n-1}, a_n] = [[a_1, \\dots , a_{n-1}], a_n]$ $(n \\ge 3)$. For $n \\ge 2$, let $T^{(n)} (A)$ be the two-sided ideal in $A$ generated by all commutators $[a_1, a_2, \\dots , a_n]$ ($a_i \\in A )$. Define $T^{(1)} (A) = A$.\n  Let $k, \\ell$ be integers such that $k > 0$, $0 \\le \\ell \\le k$. Let $m_1, \\dots , m_k$ be positive integers such that $\\ell$ of them are odd and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03136","kind":"arxiv","version":3},"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":"1610.03136","created_at":"2026-05-18T00:13:48.342512+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.03136v3","created_at":"2026-05-18T00:13:48.342512+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.03136","created_at":"2026-05-18T00:13:48.342512+00:00"},{"alias_kind":"pith_short_12","alias_value":"VVS3SWADWSTD","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VVS3SWADWSTDMLMB","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VVS3SWAD","created_at":"2026-05-18T12:30:48.956258+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/VVS3SWADWSTDMLMBY5Y5SVNLBI","json":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI.json","graph_json":"https://pith.science/api/pith-number/VVS3SWADWSTDMLMBY5Y5SVNLBI/graph.json","events_json":"https://pith.science/api/pith-number/VVS3SWADWSTDMLMBY5Y5SVNLBI/events.json","paper":"https://pith.science/paper/VVS3SWAD"},"agent_actions":{"view_html":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI","download_json":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI.json","view_paper":"https://pith.science/paper/VVS3SWAD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.03136&json=true","fetch_graph":"https://pith.science/api/pith-number/VVS3SWADWSTDMLMBY5Y5SVNLBI/graph.json","fetch_events":"https://pith.science/api/pith-number/VVS3SWADWSTDMLMBY5Y5SVNLBI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI/action/storage_attestation","attest_author":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI/action/author_attestation","sign_citation":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI/action/citation_signature","submit_replication":"https://pith.science/pith/VVS3SWADWSTDMLMBY5Y5SVNLBI/action/replication_record"}},"created_at":"2026-05-18T00:13:48.342512+00:00","updated_at":"2026-05-18T00:13:48.342512+00:00"}