{"paper":{"title":"Factorization in almost Dedekind domain","license":"http://creativecommons.org/licenses/by/4.0/","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","cross_cats":[],"primary_cat":"math.AC","authors_text":"Gyu Whan Chang, Hyun Seung Choi","submitted_at":"2026-05-17T08:13:30Z","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 "},"claims":{"count":4,"items":[{"kind":"strongest_claim","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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","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","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"572599579d8f013ceac2faa955d729c621eff553a0d23d712866de3974d886b8"},"source":{"id":"2605.17315","kind":"arxiv","version":1},"verdict":{"id":"b3981acf-5297-4cf1-903f-1e5612c76315","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:55:08.682334Z","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.","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","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.","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"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17315/integrity.json","findings":[],"available":true,"detectors_run":[{"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","findings_count":0}],"snapshot_sha256":"cd5773475dc173c9e379667cc1b7967558aaa43b6a8434df7d702a2461105243"},"references":{"count":46,"sample":[{"doi":"","year":2000,"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","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"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","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"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","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"Arnold, Krull dimension in power series rings , Trans","work_id":"a241f8af-c0dd-4a34-a92c-e5ce8c5e3485","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":", Power series rings over Pr¨ ufer domains, Pacific J. Math. 44 (1973), 1-11","work_id":"43b877d0-c7b2-40e9-b9e5-6f69d4dc5403","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":46,"snapshot_sha256":"e7b3997a1161fe49829d5a526a2219adcb821ea823d3b93168e85fdb84fbade7","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7ff697e05085374e27308e8fbfc1c7d92fa37ab2f3b6a02841d06df259c3ea2d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}