{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:IMNATUQNFBGQVAUEX66MF2RK67","short_pith_number":"pith:IMNATUQN","schema_version":"1.0","canonical_sha256":"431a09d20d284d0a8284bfbcc2ea2af7ee16c249a0777ceb1eb6f7c27c5c4093","source":{"kind":"arxiv","id":"1812.03585","version":1},"attestation_state":"computed","paper":{"title":"On some products of commutators in an associative ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexei Krasilnikov, Galina Deryabina","submitted_at":"2018-12-10T00:50:25Z","abstract_excerpt":"Let $A$ be a unital associative ring and let $T^{(k)}$ be the two-sided ideal of $A$ generated by all commutators $[a_1, a_2, \\dots , a_k]$ $(a_i \\in A)$ where $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \\dots , a_{k-1}, a_k] = \\bigl[ [a_1, \\dots , a_{k-1}], a_k \\bigr]$ $(k >2)$. It has been known that, if either $m$ or $n$ is odd then \\[ 6 \\, [a_1, a_2, \\dots , a_m] [b_1, b_2, \\dots , b_n] \\in T^{(m+n-1)} \\] for all $a_i, b_j \\in A$. This was proved by Sharma and Srivastava in 1990 and independently rediscovered later (with different proofs) by various authors. The aim of our note is to give a s"},"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":"1812.03585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-12-10T00:50:25Z","cross_cats_sorted":[],"title_canon_sha256":"8e5bcf21b7d176d7b5ba0648cb3f9748cb935049d19fa88f80677f489691be18","abstract_canon_sha256":"7943d9a58fcbc296f778365028057f365361edcce70bc8ffaaebd9b37743914f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:16.922513Z","signature_b64":"5M2HwEPwC8Ez7DuJEcHyxv7obi4xA0WMbCYoIwnyCePtJYf8ey2hsHpLeHnF8y+zEXTqA96pvycol/pPoPKVBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"431a09d20d284d0a8284bfbcc2ea2af7ee16c249a0777ceb1eb6f7c27c5c4093","last_reissued_at":"2026-05-17T23:49:16.921988Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:16.921988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some products of commutators in an associative ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexei Krasilnikov, Galina Deryabina","submitted_at":"2018-12-10T00:50:25Z","abstract_excerpt":"Let $A$ be a unital associative ring and let $T^{(k)}$ be the two-sided ideal of $A$ generated by all commutators $[a_1, a_2, \\dots , a_k]$ $(a_i \\in A)$ where $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \\dots , a_{k-1}, a_k] = \\bigl[ [a_1, \\dots , a_{k-1}], a_k \\bigr]$ $(k >2)$. It has been known that, if either $m$ or $n$ is odd then \\[ 6 \\, [a_1, a_2, \\dots , a_m] [b_1, b_2, \\dots , b_n] \\in T^{(m+n-1)} \\] for all $a_i, b_j \\in A$. This was proved by Sharma and Srivastava in 1990 and independently rediscovered later (with different proofs) by various authors. The aim of our note is to give a s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.03585","kind":"arxiv","version":1},"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":"1812.03585","created_at":"2026-05-17T23:49:16.922067+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.03585v1","created_at":"2026-05-17T23:49:16.922067+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.03585","created_at":"2026-05-17T23:49:16.922067+00:00"},{"alias_kind":"pith_short_12","alias_value":"IMNATUQNFBGQ","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_16","alias_value":"IMNATUQNFBGQVAUE","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_8","alias_value":"IMNATUQN","created_at":"2026-05-18T12:32:31.084164+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/IMNATUQNFBGQVAUEX66MF2RK67","json":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67.json","graph_json":"https://pith.science/api/pith-number/IMNATUQNFBGQVAUEX66MF2RK67/graph.json","events_json":"https://pith.science/api/pith-number/IMNATUQNFBGQVAUEX66MF2RK67/events.json","paper":"https://pith.science/paper/IMNATUQN"},"agent_actions":{"view_html":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67","download_json":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67.json","view_paper":"https://pith.science/paper/IMNATUQN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.03585&json=true","fetch_graph":"https://pith.science/api/pith-number/IMNATUQNFBGQVAUEX66MF2RK67/graph.json","fetch_events":"https://pith.science/api/pith-number/IMNATUQNFBGQVAUEX66MF2RK67/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67/action/storage_attestation","attest_author":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67/action/author_attestation","sign_citation":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67/action/citation_signature","submit_replication":"https://pith.science/pith/IMNATUQNFBGQVAUEX66MF2RK67/action/replication_record"}},"created_at":"2026-05-17T23:49:16.922067+00:00","updated_at":"2026-05-17T23:49:16.922067+00:00"}