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Over such domains summands of completely decomposable torsion-free modules are again completely decomposable (Theorem 3.2) and summands of separable torsion-free modules are likewise separable (Theorem 4.2). In addition, a Pontryagin-Hill type theorem is established on countable chains of homogeneous completely decomposable modules over h-local Pr\\\"ufer domains."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1112.0599","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-12-02T22:48:58Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"705a1bbc766cd8828dc7745f261f1611be9aba885c987372131124c601703ce7","abstract_canon_sha256":"44961ca90357f3b6f9fa4464a8c705357a4b8076b24f683b275ae173ce490c82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:06.213232Z","signature_b64":"V9pdVWtQX8/2Go+V9wu1vzOZY68XpFjPcgAQzr9yfH2CFnzKzgWIzrv8ob3qMhUBcOToL+557g9+xTP/MAl1Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fdd74da8cc0d80fd7d2d8d327dfd1dd713019dac044cf0ec6ec0710cb2e8f392","last_reissued_at":"2026-05-18T04:07:06.212508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:06.212508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On completely decomposable and separable modules over Pr\\\"ufer domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AC","authors_text":"J. 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