{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:E3BFZKSRK775CY5SFGQ32NSWUH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"805ef4e4d361d69dae6627f991077c823dabe0860e0738ec9a319237eb3756f4","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.QA","submitted_at":"2006-11-22T03:46:25Z","title_canon_sha256":"bd083111386085661fd379851576a74b5c18763ed3a65b405f09496905643f0a"},"schema_version":"1.0","source":{"id":"math/0611660","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611660","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611660v1","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611660","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"pith_short_12","alias_value":"E3BFZKSRK775","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"E3BFZKSRK775CY5S","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"E3BFZKSR","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:413befa421c855df77ac9613e5e6717cb55196a08a8821e2974ea701566a50ab","target":"graph","created_at":"2026-05-18T04:08:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce the notion of idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras, and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a ","authors_text":"Kar-Ping Shum, Siu-Hung Ng, Yuqun Chen","cross_cats":["math.RA"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2006-11-22T03:46:25Z","title":"On Radicals of Module Coalgebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611660","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1a855b51d61759e542f07addc0287f51cac34a5384e53d96584b182cb1bcfb5e","target":"record","created_at":"2026-05-18T04:08:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"805ef4e4d361d69dae6627f991077c823dabe0860e0738ec9a319237eb3756f4","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.QA","submitted_at":"2006-11-22T03:46:25Z","title_canon_sha256":"bd083111386085661fd379851576a74b5c18763ed3a65b405f09496905643f0a"},"schema_version":"1.0","source":{"id":"math/0611660","kind":"arxiv","version":1}},"canonical_sha256":"26c25caa5157ffd163b229a1bd3656a1e9bf2e09ad78ba67e3c5d258fc1a780b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26c25caa5157ffd163b229a1bd3656a1e9bf2e09ad78ba67e3c5d258fc1a780b","first_computed_at":"2026-05-18T04:08:53.135407Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:53.135407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NI0UhvZRcTrsiu9aHxShfMC/TX+0fd/c+lDWJLnGYohiTRLTzrO/Do2vOg8XveGXsiA34pDbE2Ek1BS8R6mqCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:53.135800Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0611660","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1a855b51d61759e542f07addc0287f51cac34a5384e53d96584b182cb1bcfb5e","sha256:413befa421c855df77ac9613e5e6717cb55196a08a8821e2974ea701566a50ab"],"state_sha256":"8af275414ec4331839df96d5a3d4b67c59252891dc7a1eb826468abdc249761d"}