{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:M6Q6CLSIUVWVJ67JFLKVTGEBID","short_pith_number":"pith:M6Q6CLSI","schema_version":"1.0","canonical_sha256":"67a1e12e48a56d54fbe92ad559988140f68ef01056a1fd49a86f2611c5c37dca","source":{"kind":"arxiv","id":"2201.04087","version":4},"attestation_state":"computed","paper":{"title":"Generating numbers of rings graded by amenable and supramenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Johan \\\"Oinert, Karl Lorensen","submitted_at":"2022-01-11T17:23:48Z","abstract_excerpt":"A ring $R$ has {\\it unbounded generating number} (UGN) if, for every positive integer $n$, there is no $R$-module epimorphism $R^n\\to R^{n+1}$. For a ring $R=\\bigoplus_{g\\in G} R_g$ graded by a group $G$ such that the base ring $R_1$ has UGN, we identify several sets of conditions under which $R$ must also have UGN. The most important of these are: (1) $G$ is amenable, and there is a positive integer $r$ such that, for every $g\\in G$, $R_g\\cong (R_1)^i$ as $R_1$-modules for some $i=1,\\dots,r$; (2) $G$ is supramenable, and there is a positive integer $r$ such that, for every $g\\in G$, $R_g\\cong"},"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":"2201.04087","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2022-01-11T17:23:48Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"5802a10c7e5c31e0b459126a5936fd4e26182aa9b30bc2db147fcd0bce184705","abstract_canon_sha256":"02b328d259a3f253821e290a025a018a3bb9c735a0a3981071cdff4877f020ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:50:51.111743Z","signature_b64":"bhdkM2m9lkF8X5iSD2GQ1Af+Igl/dUtoFoKmEGuYFaKQ/ASrba6NUCHZMTqzeJX/pKz+66XP14omK2tVCukbAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67a1e12e48a56d54fbe92ad559988140f68ef01056a1fd49a86f2611c5c37dca","last_reissued_at":"2026-07-05T06:50:51.111267Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:50:51.111267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generating numbers of rings graded by amenable and supramenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Johan \\\"Oinert, Karl Lorensen","submitted_at":"2022-01-11T17:23:48Z","abstract_excerpt":"A ring $R$ has {\\it unbounded generating number} (UGN) if, for every positive integer $n$, there is no $R$-module epimorphism $R^n\\to R^{n+1}$. For a ring $R=\\bigoplus_{g\\in G} R_g$ graded by a group $G$ such that the base ring $R_1$ has UGN, we identify several sets of conditions under which $R$ must also have UGN. The most important of these are: (1) $G$ is amenable, and there is a positive integer $r$ such that, for every $g\\in G$, $R_g\\cong (R_1)^i$ as $R_1$-modules for some $i=1,\\dots,r$; (2) $G$ is supramenable, and there is a positive integer $r$ such that, for every $g\\in G$, $R_g\\cong"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.04087","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.04087/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2201.04087","created_at":"2026-07-05T06:50:51.111343+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.04087v4","created_at":"2026-07-05T06:50:51.111343+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.04087","created_at":"2026-07-05T06:50:51.111343+00:00"},{"alias_kind":"pith_short_12","alias_value":"M6Q6CLSIUVWV","created_at":"2026-07-05T06:50:51.111343+00:00"},{"alias_kind":"pith_short_16","alias_value":"M6Q6CLSIUVWVJ67J","created_at":"2026-07-05T06:50:51.111343+00:00"},{"alias_kind":"pith_short_8","alias_value":"M6Q6CLSI","created_at":"2026-07-05T06:50:51.111343+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/M6Q6CLSIUVWVJ67JFLKVTGEBID","json":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID.json","graph_json":"https://pith.science/api/pith-number/M6Q6CLSIUVWVJ67JFLKVTGEBID/graph.json","events_json":"https://pith.science/api/pith-number/M6Q6CLSIUVWVJ67JFLKVTGEBID/events.json","paper":"https://pith.science/paper/M6Q6CLSI"},"agent_actions":{"view_html":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID","download_json":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID.json","view_paper":"https://pith.science/paper/M6Q6CLSI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.04087&json=true","fetch_graph":"https://pith.science/api/pith-number/M6Q6CLSIUVWVJ67JFLKVTGEBID/graph.json","fetch_events":"https://pith.science/api/pith-number/M6Q6CLSIUVWVJ67JFLKVTGEBID/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID/action/storage_attestation","attest_author":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID/action/author_attestation","sign_citation":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID/action/citation_signature","submit_replication":"https://pith.science/pith/M6Q6CLSIUVWVJ67JFLKVTGEBID/action/replication_record"}},"created_at":"2026-07-05T06:50:51.111343+00:00","updated_at":"2026-07-05T06:50:51.111343+00:00"}