{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XQZPB7QJ6QQKQMTXUJUVY4B2MO","short_pith_number":"pith:XQZPB7QJ","canonical_record":{"source":{"id":"1210.2893","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T12:47:35Z","cross_cats_sorted":[],"title_canon_sha256":"4680dcfba87c17461282c6986de8505504cb66bebb59700ffd74d810a8906d80","abstract_canon_sha256":"52516016ef73d29d845fc16474530eee3199b6f7adee311a46652a760d14e1ea"},"schema_version":"1.0"},"canonical_sha256":"bc32f0fe09f420a83277a2695c703a63b4638d382dd60649d30cd0a2e901c792","source":{"kind":"arxiv","id":"1210.2893","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.2893","created_at":"2026-05-18T01:03:37Z"},{"alias_kind":"arxiv_version","alias_value":"1210.2893v1","created_at":"2026-05-18T01:03:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2893","created_at":"2026-05-18T01:03:37Z"},{"alias_kind":"pith_short_12","alias_value":"XQZPB7QJ6QQK","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XQZPB7QJ6QQKQMTX","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XQZPB7QJ","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XQZPB7QJ6QQKQMTXUJUVY4B2MO","target":"record","payload":{"canonical_record":{"source":{"id":"1210.2893","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T12:47:35Z","cross_cats_sorted":[],"title_canon_sha256":"4680dcfba87c17461282c6986de8505504cb66bebb59700ffd74d810a8906d80","abstract_canon_sha256":"52516016ef73d29d845fc16474530eee3199b6f7adee311a46652a760d14e1ea"},"schema_version":"1.0"},"canonical_sha256":"bc32f0fe09f420a83277a2695c703a63b4638d382dd60649d30cd0a2e901c792","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:37.304577Z","signature_b64":"2oBCnmrp40ywSp0T2Yo/PSvG/Vq9YJ5T/z64R3L0sSw2zOcM4p3dUpe9pJ7K/qgyDwTEw+lN8jALkESgNbCxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc32f0fe09f420a83277a2695c703a63b4638d382dd60649d30cd0a2e901c792","last_reissued_at":"2026-05-18T01:03:37.303856Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:37.303856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.2893","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8L7SgjOXrv4RDomBODSYjy7eb3tjc4pBHW2O/mQlBlfRJ8hy14Rp15bqhW/wE42WCRZIlgDLOwZXj8CxZ+mhAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T14:01:05.049743Z"},"content_sha256":"fab08eead22d7593490f387958655961d5ba101e3414ea10dfc484af6b1f5513","schema_version":"1.0","event_id":"sha256:fab08eead22d7593490f387958655961d5ba101e3414ea10dfc484af6b1f5513"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XQZPB7QJ6QQKQMTXUJUVY4B2MO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rad-supplementing modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Salahattin \\\"Ozdemir","submitted_at":"2012-10-10T12:47:35Z","abstract_excerpt":"Let R be an associative ring with unity and let M be an R-module. We call M (ample) Rad-supplementing if M has a (ample) Rad-supplement in every extension. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal and reduced, and the direct sum of countably many copies of R is Rad-supplementing if and only if R is re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2893","kind":"arxiv","version":1},"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":"2026-05-18T01:03:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"heK9FmCLF+g/h7TZW57EsLiAt6EhR8cnkf+zaTb9GJAOonaR0TOYH60qj4DbYccY3FpsQTrcXhUIPLlEjna1Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T14:01:05.050090Z"},"content_sha256":"191d7d15535fde230af45db0a039d12bc6142ea5276716f3b947d4c0906e1f0e","schema_version":"1.0","event_id":"sha256:191d7d15535fde230af45db0a039d12bc6142ea5276716f3b947d4c0906e1f0e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XQZPB7QJ6QQKQMTXUJUVY4B2MO/bundle.json","state_url":"https://pith.science/pith/XQZPB7QJ6QQKQMTXUJUVY4B2MO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XQZPB7QJ6QQKQMTXUJUVY4B2MO/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":"2026-06-26T14:01:05Z","links":{"resolver":"https://pith.science/pith/XQZPB7QJ6QQKQMTXUJUVY4B2MO","bundle":"https://pith.science/pith/XQZPB7QJ6QQKQMTXUJUVY4B2MO/bundle.json","state":"https://pith.science/pith/XQZPB7QJ6QQKQMTXUJUVY4B2MO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XQZPB7QJ6QQKQMTXUJUVY4B2MO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XQZPB7QJ6QQKQMTXUJUVY4B2MO","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":"52516016ef73d29d845fc16474530eee3199b6f7adee311a46652a760d14e1ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T12:47:35Z","title_canon_sha256":"4680dcfba87c17461282c6986de8505504cb66bebb59700ffd74d810a8906d80"},"schema_version":"1.0","source":{"id":"1210.2893","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.2893","created_at":"2026-05-18T01:03:37Z"},{"alias_kind":"arxiv_version","alias_value":"1210.2893v1","created_at":"2026-05-18T01:03:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2893","created_at":"2026-05-18T01:03:37Z"},{"alias_kind":"pith_short_12","alias_value":"XQZPB7QJ6QQK","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XQZPB7QJ6QQKQMTX","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XQZPB7QJ","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:191d7d15535fde230af45db0a039d12bc6142ea5276716f3b947d4c0906e1f0e","target":"graph","created_at":"2026-05-18T01:03:37Z","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 R be an associative ring with unity and let M be an R-module. We call M (ample) Rad-supplementing if M has a (ample) Rad-supplement in every extension. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal and reduced, and the direct sum of countably many copies of R is Rad-supplementing if and only if R is re","authors_text":"Salahattin \\\"Ozdemir","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T12:47:35Z","title":"Rad-supplementing modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2893","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:fab08eead22d7593490f387958655961d5ba101e3414ea10dfc484af6b1f5513","target":"record","created_at":"2026-05-18T01:03:37Z","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":"52516016ef73d29d845fc16474530eee3199b6f7adee311a46652a760d14e1ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T12:47:35Z","title_canon_sha256":"4680dcfba87c17461282c6986de8505504cb66bebb59700ffd74d810a8906d80"},"schema_version":"1.0","source":{"id":"1210.2893","kind":"arxiv","version":1}},"canonical_sha256":"bc32f0fe09f420a83277a2695c703a63b4638d382dd60649d30cd0a2e901c792","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc32f0fe09f420a83277a2695c703a63b4638d382dd60649d30cd0a2e901c792","first_computed_at":"2026-05-18T01:03:37.303856Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:37.303856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2oBCnmrp40ywSp0T2Yo/PSvG/Vq9YJ5T/z64R3L0sSw2zOcM4p3dUpe9pJ7K/qgyDwTEw+lN8jALkESgNbCxDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:37.304577Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.2893","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fab08eead22d7593490f387958655961d5ba101e3414ea10dfc484af6b1f5513","sha256:191d7d15535fde230af45db0a039d12bc6142ea5276716f3b947d4c0906e1f0e"],"state_sha256":"e223290e92536dc8628a263ea9fd25f80cac6cc64b80ba911b00b903efe674ca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DA4aZCS68kZg1Ftby98M+Ygu5PqcrNaiMVbRownJ1lpwLWpqunzdiDHM0C1aDDiYKmtH0LnFzKFQ0yCxJz9IDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T14:01:05.052057Z","bundle_sha256":"c3f7227a0e3dd0d7b617d5e71590ae60ef7f7e5280215ba8cb881f9581781f92"}}