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We show that in an order $R$ of a Dedekind domain every regular prime ideal can be generated by an idempotent pair; moreover, if $R$ is PRINC, then its integral closure, which is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and only if it is a PID. Furthermore, we show that the only imaginary quadratic orders $\\mathbb Z[\\sqrt{-d}]$, $d > 0$ square-free, "},"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":"1412.8089","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-12-27T23:09:03Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"c8b8c7f552cea75b66f9944c94e29b2ddc7ce4c5871f4849611a9bb44006c97e","abstract_canon_sha256":"6c34dde168909769c28907216f84e5fdacb816627748ad573cd09e1db6087026"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:24.492845Z","signature_b64":"x9Xb8ZwRRV/wpMFC6Em0mjOc1F6FwlXqV62zYVE5GqLNnUD++Wt02y+g0ivRkfIJGal0Wl8NJPzS8HlCSgQTCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a581ec432da1e7e97552a220b7beaf6cda928cc96ca013d6f96e95f27c973174","last_reissued_at":"2026-05-18T00:04:24.492015Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:24.492015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Idempotent pairs and PRINC domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.RA","authors_text":"Giulio Peruginelli, Luigi Salce, Paolo Zanardo","submitted_at":"2014-12-27T23:09:03Z","abstract_excerpt":"A pair of elements $a,b$ in an integral domain $R$ is an idempotent pair if either $a(1-a) \\in bR$, or $b(1-b) \\in aR$. $R$ is said to be a PRINC domain if all the ideals generated by an idempotent pair are principal. We show that in an order $R$ of a Dedekind domain every regular prime ideal can be generated by an idempotent pair; moreover, if $R$ is PRINC, then its integral closure, which is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and only if it is a PID. 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