{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:GC375GL7AZOSTEXRTCN6O2N4O5","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":"1c4f1db87d28214086a3d61971d8b87830ec91c478692e9bc6ddcd4abc862cde","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.QA","submitted_at":"2003-11-28T09:25:41Z","title_canon_sha256":"ad2824bba74d0105aeed2cc6b2bdcfbf4c0eb216b08c60d7b255fb68263b5bbb"},"schema_version":"1.0","source":{"id":"math/0311521","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0311521","created_at":"2026-07-04T14:38:02Z"},{"alias_kind":"arxiv_version","alias_value":"math/0311521v1","created_at":"2026-07-04T14:38:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0311521","created_at":"2026-07-04T14:38:02Z"},{"alias_kind":"pith_short_12","alias_value":"GC375GL7AZOS","created_at":"2026-07-04T14:38:02Z"},{"alias_kind":"pith_short_16","alias_value":"GC375GL7AZOSTEXR","created_at":"2026-07-04T14:38:02Z"},{"alias_kind":"pith_short_8","alias_value":"GC375GL7","created_at":"2026-07-04T14:38:02Z"}],"graph_snapshots":[{"event_id":"sha256:32dce2a9d7c3ad34bd3bbef2f5fa4b27596592e30b2d989ae921d19988198736","target":"graph","created_at":"2026-07-04T14:38:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0311521/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"T. Shudo and H. Miyamito \\cite{SM78} showed that $C$ can be decomposed into a direct sum of its indecomposable subcoalgebras of $C$.\n  Y.H. Xu \\cite {XF92} showed that the decomposition was unique. He also showed that $M$ can uniquely be decomposed into a direct sum of the weak-closed indecomposable subcomodules of $M$(we call the decomposition the weak-closed indecomposable decomposition) in \\cite{XSF94}. In this paper, we give the relation between the two decomposition. We show that if $M$ is a full, $W$-relational hereditary $C$-comodule, then the following conclusions hold: (1) $M$ is inde","authors_text":"Shouchuan Zhang","cross_cats":["math.RA"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2003-11-28T09:25:41Z","title":"The relation between the decomposition of comodules and coalgebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0311521","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":"sha256:935537c0dc0be69cd961d79905b1989f7337daeba5f205799deedc033d505d68","target":"record","created_at":"2026-07-04T14:38:02Z","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":"1c4f1db87d28214086a3d61971d8b87830ec91c478692e9bc6ddcd4abc862cde","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.QA","submitted_at":"2003-11-28T09:25:41Z","title_canon_sha256":"ad2824bba74d0105aeed2cc6b2bdcfbf4c0eb216b08c60d7b255fb68263b5bbb"},"schema_version":"1.0","source":{"id":"math/0311521","kind":"arxiv","version":1}},"canonical_sha256":"30b7fe997f065d2992f1989be769bc7748655e9a078c06087e92a8f32462c637","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"30b7fe997f065d2992f1989be769bc7748655e9a078c06087e92a8f32462c637","first_computed_at":"2026-07-04T14:38:02.387734Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:38:02.387734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FNkdp5GQVdF5vl0WRI/HbTo4bk5DUFw1YDgDNo1J0cyLz5Z5gBhW+92zj8vCUXOJwti4gUUbtRrMkyeb2/YlAQ==","signature_status":"signed_v1","signed_at":"2026-07-04T14:38:02.388097Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0311521","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:935537c0dc0be69cd961d79905b1989f7337daeba5f205799deedc033d505d68","sha256:32dce2a9d7c3ad34bd3bbef2f5fa4b27596592e30b2d989ae921d19988198736"],"state_sha256":"cec8bfb3fed3b6a46f500d34c9826c613c0ff8e5b0c6a7a5fa824c0cb4035a39"}