{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TX3ZUXDWGYNU4IUAXPLHGXMSL7","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":"8f67b635e447b2d5899465d6ea83a4ce7625ab152a855c3484080ea6fe1b0fc7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-17T08:13:30Z","title_canon_sha256":"3531ebc24a3757fb6af382e2d5bc506a0d775154b69a8a6022b2d3fde7ad1ecd"},"schema_version":"1.0","source":{"id":"2605.17315","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.17315","created_at":"2026-05-20T00:03:51Z"},{"alias_kind":"arxiv_version","alias_value":"2605.17315v1","created_at":"2026-05-20T00:03:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17315","created_at":"2026-05-20T00:03:51Z"},{"alias_kind":"pith_short_12","alias_value":"TX3ZUXDWGYNU","created_at":"2026-05-20T00:03:51Z"},{"alias_kind":"pith_short_16","alias_value":"TX3ZUXDWGYNU4IUA","created_at":"2026-05-20T00:03:51Z"},{"alias_kind":"pith_short_8","alias_value":"TX3ZUXDW","created_at":"2026-05-20T00:03:51Z"}],"graph_snapshots":[{"event_id":"sha256:7d7bc8c0e1ed3f977ae07f682965b55f5a16580e3c533afaa8fd3e07a7da74d6","target":"graph","created_at":"2026-05-20T00:03:51Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"If F is algebraically closed or a finite field of char(F)=p, then D has no irreducible element. We also show that if F=Q and p=2, every nonzero nonunit of D can be written as a product of countably many prime elements of D and every proper nonzero principal ideal of D can be uniquely written as a countable intersection of principal primary ideals."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The ring D is almost Dedekind precisely when char(F) ≠ p, and the irreducibility criteria rely on properties of cyclotomic polynomials and field extensions in the specific construction of the D_n."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"In the almost Dedekind domain D built from field F and prime p, the paper gives conditions for irreducibles in D_n to stay irreducible in D, shows D has no irreducibles for algebraically closed F or finite F of characteristic p, and for F=Q and p=2 proves every nonzero nonunit factors into countably"},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"In the ring D formed as the union of F-adjoined p-power roots of X and their inverses, there are no irreducible elements when F is algebraically closed or finite of characteristic p, while countable prime factorizations exist for F equal to"}],"snapshot_sha256":"572599579d8f013ceac2faa955d729c621eff553a0d23d712866de3974d886b8"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7ff697e05085374e27308e8fbfc1c7d92fa37ab2f3b6a02841d06df259c3ea2d"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T23:01:52.375777Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.686888Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.782231Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.752039Z","status":"skipped","version":"1.0.0"}],"endpoint":"/pith/2605.17315/integrity.json","findings":[],"snapshot_sha256":"cd5773475dc173c9e379667cc1b7967558aaa43b6a8434df7d702a2461105243","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $F$ be a field, $p$ a prime number, $X$ an indeterminate over $F$, $D_n =F[X^{\\frac{1}{p^n}}, X^{-\\frac{1}{p^n}}]$ for each integer $n \\geq 0$ and $D = \\bigcup\\limits_{n\\in\\mathbb{N}_0}D_n.$ Then $D$ is a one-dimensional B{\\'e}zout domain but not a Dedekind domain, and $D$ is an almost Dedekind domain if and only if char$(F) \\neq p$. In this paper, we study the element-wise factorization properties of $D$. For example, we determine when an irreducible element of $D_n$ is an irreducible element of $D$, in terms of $n$ and $p$. In particular, we show that if $F$ is algebraically closed or a ","authors_text":"Gyu Whan Chang, Hyun Seung Choi","cross_cats":[],"headline":"In the ring D formed as the union of F-adjoined p-power roots of X and their inverses, there are no irreducible elements when F is algebraically closed or finite of characteristic p, while countable prime factorizations exist for F equal to","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-17T08:13:30Z","title":"Factorization in almost Dedekind domain"},"references":{"count":46,"internal_anchors":0,"resolved_work":46,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"D. D. Anderson, GCD Domains, Gauss’ Lemma, and Contents of Polynomials , Non- Noetherian Commutative Ring Theory, Mathematics and Its Applications 520 (Kluwer Aca- demic Publishers, Dordrecht, 2000) p","work_id":"f8d6748c-74c3-4091-a50e-15b34283cb12","year":2000},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"D. D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility , Proc. Amer. Math. Soc. 109(4) (1990), 907-913","work_id":"ff84a80c-eb23-4ec6-ac27-fc958d188bf6","year":1990},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"D. D. Anderson and M. Zafrullah, A generalization of unique factorization , Bollettino U.M.I. 9-A (1995), 401-413","work_id":"76edabc4-5d56-47ac-a2c5-13faf99e3ac7","year":1995},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Arnold, Krull dimension in power series rings , Trans","work_id":"a241f8af-c0dd-4a34-a92c-e5ce8c5e3485","year":1973},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":", Power series rings over Pr¨ ufer domains, Pacific J. Math. 44 (1973), 1-11","work_id":"43b877d0-c7b2-40e9-b9e5-6f69d4dc5403","year":1973}],"snapshot_sha256":"e7b3997a1161fe49829d5a526a2219adcb821ea823d3b93168e85fdb84fbade7"},"source":{"id":"2605.17315","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T22:55:08.682334Z","id":"b3981acf-5297-4cf1-903f-1e5612c76315","model_set":{"reader":"grok-4.3"},"one_line_summary":"In the almost Dedekind domain D built from field F and prime p, the paper gives conditions for irreducibles in D_n to stay irreducible in D, shows D has no irreducibles for algebraically closed F or finite F of characteristic p, and for F=Q and p=2 proves every nonzero nonunit factors into countably","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"In the ring D formed as the union of F-adjoined p-power roots of X and their inverses, there are no irreducible elements when F is algebraically closed or finite of characteristic p, while countable prime factorizations exist for F equal to","strongest_claim":"If F is algebraically closed or a finite field of char(F)=p, then D has no irreducible element. We also show that if F=Q and p=2, every nonzero nonunit of D can be written as a product of countably many prime elements of D and every proper nonzero principal ideal of D can be uniquely written as a countable intersection of principal primary ideals.","weakest_assumption":"The ring D is almost Dedekind precisely when char(F) ≠ p, and the irreducibility criteria rely on properties of cyclotomic polynomials and field extensions in the specific construction of the D_n."}},"verdict_id":"b3981acf-5297-4cf1-903f-1e5612c76315"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:91047faff3a7964977fab16b96e8de2235a95387f6229c5d81788dc8bb9791d0","target":"record","created_at":"2026-05-20T00:03:51Z","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":"8f67b635e447b2d5899465d6ea83a4ce7625ab152a855c3484080ea6fe1b0fc7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-17T08:13:30Z","title_canon_sha256":"3531ebc24a3757fb6af382e2d5bc506a0d775154b69a8a6022b2d3fde7ad1ecd"},"schema_version":"1.0","source":{"id":"2605.17315","kind":"arxiv","version":1}},"canonical_sha256":"9df79a5c76361b4e2280bbd6735d925fefb064eac36be012511f05778e97fbb9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9df79a5c76361b4e2280bbd6735d925fefb064eac36be012511f05778e97fbb9","first_computed_at":"2026-05-20T00:03:51.726169Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:51.726169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CMgWzBUxF+vXcmplUv8pddiU6OZNf2A7i8QZiuKnfT2D+XPu8h/6j8eCE91b4Ju9ihhOOihzSel4GDF8EKSeBA==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:51.727153Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17315","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91047faff3a7964977fab16b96e8de2235a95387f6229c5d81788dc8bb9791d0","sha256:7d7bc8c0e1ed3f977ae07f682965b55f5a16580e3c533afaa8fd3e07a7da74d6"],"state_sha256":"6633cdab3671c4961fa4565257cba82c6cf34cd789030c298c43b6d95f40d7c1"}