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Beyond the Cohen--Macaulay property, the existence of such modules forces further conditions on the ring, including reducedness, normality, being an integral domain, and various conditions such as being complete intersection or Gorenstein.\n  We also address the reflexivity and torsionlessness of modules of finite injective dimension, showing that these properties force the ring to be quasi-normal. In the same vein, we investigate th","authors_text":"Mohsen Asgharzadeh","cross_cats":[],"headline":"Existence of finitely generated modules of finite injective dimension forces the ring to be reduced, normal, an integral domain, complete intersection or Gorenstein beyond Cohen-Macaulay.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2023-01-03T13:59:12Z","title":"Notes on modules of finite injective dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.01105","kind":"arxiv","version":6},"verdict":{"created_at":"2026-05-24T10:09:31.467082Z","id":"b8619f19-1371-4574-b6d7-60099106e7dc","model_set":{"reader":"grok-4.3"},"one_line_summary":"Finitely generated modules of finite injective dimension over a ring force the ring to satisfy properties including reducedness, normality, being an integral domain, complete intersection, or Gorenstein.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Existence of finitely generated modules of finite injective dimension forces the ring to be reduced, normal, an integral domain, complete intersection or Gorenstein beyond Cohen-Macaulay.","strongest_claim":"The existence of finitely generated modules of finite injective dimension forces the ambient ring to be reduced, normal, an integral domain, complete intersection, or Gorenstein, beyond the Cohen-Macaulay property.","weakest_assumption":"The paper assumes the standard Noetherian commutative ring setting of the Bass conjecture and that the modules under study are finitely generated over that ring; if the ring is not Noetherian or the modules are not finitely generated, the forcing statements on ring properties may fail to hold."}},"verdict_id":"b8619f19-1371-4574-b6d7-60099106e7dc"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ce30c01c2222b3727d718cd6779465f585beece23c97ad2c1bf2ab3f11b01d95","target":"record","created_at":"2026-05-26T02:04:58Z","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":"57be155bccc5b19205157c00264bf96e9464f28d1cdf62536f7f22e27d85e4bf","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2023-01-03T13:59:12Z","title_canon_sha256":"c5ffb98a8c28d2fb197a73d6d63efa0326d938ebbb5293ef9311dbb911931962"},"schema_version":"1.0","source":{"id":"2301.01105","kind":"arxiv","version":6}},"canonical_sha256":"7fa44908d94405579e0a445b664983899a07509ccd5e79be386474fe6245cbfe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7fa44908d94405579e0a445b664983899a07509ccd5e79be386474fe6245cbfe","first_computed_at":"2026-05-26T02:04:58.013441Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T02:04:58.013441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A//W23/mhphL/ttjiI2YUgAS1dpeKvuOFkkCDQ6UVqIsMTs/TZr+75WRkTUrElbxSzP9MUO0ZLyTkBybAGgjBg==","signature_status":"signed_v1","signed_at":"2026-05-26T02:04:58.014065Z","signed_message":"canonical_sha256_bytes"},"source_id":"2301.01105","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce30c01c2222b3727d718cd6779465f585beece23c97ad2c1bf2ab3f11b01d95","sha256:52ed82b10e56b50b3eadeb17f5ba60189e84d73d002101ef4b99b838b1f26a9a"],"state_sha256":"bb408302423fcdae7223390cd51f91d37cfd12bdd10d44b04611321e1e58e2f0"}