{"paper":{"title":"When every principal ideal is flat","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Fatima Cheniour, Najib Mahdou","submitted_at":"2010-12-12T13:16:20Z","abstract_excerpt":"This paper deals with well-known notion of $PF$-rings, that is, rings in which principal ideals are flat.\n  We give a new characterization of $PF$-rings. Also, we provide a necessary and sufficient condition for $R\\bowtie I$ (resp., $R/I$ when $R$ is a Dedekind domain or $I$ is a primary ideal) to be $PF$-ring. The article includes a brief discussion of the scope and precision of our results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2538","kind":"arxiv","version":2},"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"}