{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:52TPIXELQT5WHSEGQETWAQV7XP","short_pith_number":"pith:52TPIXEL","canonical_record":{"source":{"id":"1309.0409","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.RA","submitted_at":"2013-09-02T14:00:28Z","cross_cats_sorted":[],"title_canon_sha256":"411da039263f871c1e85a35e12a856832570b64a84f03c4b1d073ea4fd4bac73","abstract_canon_sha256":"abadf4b47c762c6f10c60fdf8f28c0e1a3b5ccfece79c811a8e9a654c7fb40bb"},"schema_version":"1.0"},"canonical_sha256":"eea6f45c8b84fb63c88681276042bfbbf495bc59825a2e43c26771080e0ebebc","source":{"kind":"arxiv","id":"1309.0409","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0409","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0409v4","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0409","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"pith_short_12","alias_value":"52TPIXELQT5W","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"52TPIXELQT5WHSEG","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"52TPIXEL","created_at":"2026-05-18T12:27:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:52TPIXELQT5WHSEGQETWAQV7XP","target":"record","payload":{"canonical_record":{"source":{"id":"1309.0409","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.RA","submitted_at":"2013-09-02T14:00:28Z","cross_cats_sorted":[],"title_canon_sha256":"411da039263f871c1e85a35e12a856832570b64a84f03c4b1d073ea4fd4bac73","abstract_canon_sha256":"abadf4b47c762c6f10c60fdf8f28c0e1a3b5ccfece79c811a8e9a654c7fb40bb"},"schema_version":"1.0"},"canonical_sha256":"eea6f45c8b84fb63c88681276042bfbbf495bc59825a2e43c26771080e0ebebc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:03.601161Z","signature_b64":"iHL7SA3hgutV8AitmiCayY/uEkzsNZAGBKQgiEkryvO+qcA7btSqUAgiCQv6e9Q2ie38NB9A1w6fyGxTglU4Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eea6f45c8b84fb63c88681276042bfbbf495bc59825a2e43c26771080e0ebebc","last_reissued_at":"2026-05-18T03:07:03.600775Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:03.600775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.0409","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"reVrf1oIV8zVoEmni6hfHFQf+5A9ybpUMGOQPKqVPUJxycQFhiEqtxLElgQMWyepiOBpeKp2yjLGU2VvxkDhAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:48:37.759957Z"},"content_sha256":"45757f1f8a8e2bf2b18e7a94926bcea688e19d3e569d52cdaec346efd5cfbd27","schema_version":"1.0","event_id":"sha256:45757f1f8a8e2bf2b18e7a94926bcea688e19d3e569d52cdaec346efd5cfbd27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:52TPIXELQT5WHSEGQETWAQV7XP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The endomorphism ring of an injective square-free module","license":"http://creativecommons.org/licenses/publicdomain/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Mai Hoang Bien","submitted_at":"2013-09-02T14:00:28Z","abstract_excerpt":"Let $M_R$ be an injective right module over a ring $R$. The goal of this paper to prove that the endomorphism ring $S=(M_R)$ of $M$ is quasi-duo if and only if $M_R$ is square-free."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0409","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kPrlpfNOvAI5zyQ7NfrupUJsSfqqg+8CbTLLO7dZMbYu7kzqveNc6D9aRcR5yWtv2X0sg+kRGVlrjA7rYldaDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:48:37.760289Z"},"content_sha256":"f69f8553dcc7a03326bee4f037762143b033911ace4466741aa14bff764e6030","schema_version":"1.0","event_id":"sha256:f69f8553dcc7a03326bee4f037762143b033911ace4466741aa14bff764e6030"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/52TPIXELQT5WHSEGQETWAQV7XP/bundle.json","state_url":"https://pith.science/pith/52TPIXELQT5WHSEGQETWAQV7XP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/52TPIXELQT5WHSEGQETWAQV7XP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T04:48:37Z","links":{"resolver":"https://pith.science/pith/52TPIXELQT5WHSEGQETWAQV7XP","bundle":"https://pith.science/pith/52TPIXELQT5WHSEGQETWAQV7XP/bundle.json","state":"https://pith.science/pith/52TPIXELQT5WHSEGQETWAQV7XP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/52TPIXELQT5WHSEGQETWAQV7XP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:52TPIXELQT5WHSEGQETWAQV7XP","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":"abadf4b47c762c6f10c60fdf8f28c0e1a3b5ccfece79c811a8e9a654c7fb40bb","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.RA","submitted_at":"2013-09-02T14:00:28Z","title_canon_sha256":"411da039263f871c1e85a35e12a856832570b64a84f03c4b1d073ea4fd4bac73"},"schema_version":"1.0","source":{"id":"1309.0409","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0409","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0409v4","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0409","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"pith_short_12","alias_value":"52TPIXELQT5W","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"52TPIXELQT5WHSEG","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"52TPIXEL","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:f69f8553dcc7a03326bee4f037762143b033911ace4466741aa14bff764e6030","target":"graph","created_at":"2026-05-18T03:07:03Z","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":"Let $M_R$ be an injective right module over a ring $R$. The goal of this paper to prove that the endomorphism ring $S=(M_R)$ of $M$ is quasi-duo if and only if $M_R$ is square-free.","authors_text":"Mai Hoang Bien","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.RA","submitted_at":"2013-09-02T14:00:28Z","title":"The endomorphism ring of an injective square-free module"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0409","kind":"arxiv","version":4},"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":"sha256:45757f1f8a8e2bf2b18e7a94926bcea688e19d3e569d52cdaec346efd5cfbd27","target":"record","created_at":"2026-05-18T03:07:03Z","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":"abadf4b47c762c6f10c60fdf8f28c0e1a3b5ccfece79c811a8e9a654c7fb40bb","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.RA","submitted_at":"2013-09-02T14:00:28Z","title_canon_sha256":"411da039263f871c1e85a35e12a856832570b64a84f03c4b1d073ea4fd4bac73"},"schema_version":"1.0","source":{"id":"1309.0409","kind":"arxiv","version":4}},"canonical_sha256":"eea6f45c8b84fb63c88681276042bfbbf495bc59825a2e43c26771080e0ebebc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eea6f45c8b84fb63c88681276042bfbbf495bc59825a2e43c26771080e0ebebc","first_computed_at":"2026-05-18T03:07:03.600775Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:03.600775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iHL7SA3hgutV8AitmiCayY/uEkzsNZAGBKQgiEkryvO+qcA7btSqUAgiCQv6e9Q2ie38NB9A1w6fyGxTglU4Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:03.601161Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.0409","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:45757f1f8a8e2bf2b18e7a94926bcea688e19d3e569d52cdaec346efd5cfbd27","sha256:f69f8553dcc7a03326bee4f037762143b033911ace4466741aa14bff764e6030"],"state_sha256":"acc63bde6784726b868c4e2578e8f0a5bd49190531ca0a3ebbc5040687106dd3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bBy/wM3LrLfbvQcq99/oHYXaB3uaVH9IV+abJ4TMtAim4REjdxiz/H54N8eT+UxSM85pIwPHXRGrpduo/zexDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T04:48:37.762446Z","bundle_sha256":"379e7b9c34f24c1dc205b9bbb67e75fbce2795f4bab0a8467d0f8c1a431c5d1a"}}