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It can be easily seen that the additive group of the quotient ring Z<X> /T^(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z<X> /T^(3) is free abelian as well. In the present note we show that this is not the cas"},"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":"1204.2674","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-04-12T10:20:06Z","cross_cats_sorted":[],"title_canon_sha256":"b124364f33332a0b4691d8d2feafceaab01f35db49e5b389213fa8e4a0e3320c","abstract_canon_sha256":"b9e36b7b16834dfc4b79ce04f2a8ea338b252215b7d9a7e33228deeca8a1ff4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:38.537212Z","signature_b64":"YjSih7UJCYka688xZADpAh7VNh0PFviFDMSMUSK0nIT62HbXqOUu9rGROjyZTsh12EO+MYPp/teVK9S3wOFIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc52f36870519c342eb2a86b4f9df5e9197af547248d41d8850fc31e5c2d5485","last_reissued_at":"2026-05-18T03:08:38.536647Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:38.536647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The additive group of a Lie nilpotent associative ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexei Krasilnikov","submitted_at":"2012-04-12T10:20:06Z","abstract_excerpt":"Let Z<X> be the free unitary associative ring freely generated by an infinite countable set X = {x_1, x_2,...}. Define a left-normed commutator [x_1, x_2, ..., x_n] by [a,b] = ab - ba, [a,b,c] = [[a,b],c]. For n \\ge 2, let T^(n) be the ideal in Z<X> generated by all commutators [a_1,a_2,..., a_n] (a_i \\in Z<X>). It can be easily seen that the additive group of the quotient ring Z<X> /T^(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z<X> /T^(3) is free abelian as well. 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