{"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"}