{"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. Furthermore, we show that the only imaginary quadratic orders $\\mathbb Z[\\sqrt{-d}]$, $d > 0$ square-free, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8089","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}