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For subgroups $H$ of $G$, left ideals $J$ of $R(H)$ and right $H$-submodules $M$ of $I_Z(G)$ the quotients $I_R(G)J/MJ$ are studied by homological methods, notably for $M= I_Z(G)I_Z(H)$, $I_Z(H)I_Z(G) + I_Z([H,G])Z(G)$ and $Z(G)I_Z(N) +I^n_{Z,\\cal G}(G)$ with $N \\lhd G$ where the group $I_R(G)J/MJ$ is completely determined for $n=2$. The groups $I^{n-1}_{Z,\\cal G}(G)I_Z(H)/I^n_{Z,\\cal G}"},"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":"0707.0281","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2007-07-02T18:58:03Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"3568428b45a16df9c3271683f2f6e76dba9ea5512734f5827afc049187213150","abstract_canon_sha256":"ac0f482277de91351865fce829a547a3e4a4945718b9db23b5f0fb66c66d7ef6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:35.116947Z","signature_b64":"Iz5ITIusTYj3vMQkj3JIl4t186wlB4e2zMRc4l/TZB7xpJGAmuhFR+w62RzAxnvrfbq5AZH9zdOyfnoUJsavDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1d7e57c5804cb5120c35764eeba18acc1432170202bdb79115bf645a52a234f","last_reissued_at":"2026-05-18T04:18:35.116408Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:35.116408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Fox quotients of arbitrary group algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Manfred Hartl","submitted_at":"2007-07-02T18:58:03Z","abstract_excerpt":"For a group $G$, N-series $\\cal G$ of $G$ and commutative ring $R$ let $I^n_{R,\\cal G}(G)$, $n\\ge 0$, denote the filtration of the group algebra $R(G)$ induced by $\\cal G$, and $I_R(G)$ its augmentation ideal. For subgroups $H$ of $G$, left ideals $J$ of $R(H)$ and right $H$-submodules $M$ of $I_Z(G)$ the quotients $I_R(G)J/MJ$ are studied by homological methods, notably for $M= I_Z(G)I_Z(H)$, $I_Z(H)I_Z(G) + I_Z([H,G])Z(G)$ and $Z(G)I_Z(N) +I^n_{Z,\\cal G}(G)$ with $N \\lhd G$ where the group $I_R(G)J/MJ$ is completely determined for $n=2$. 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