{"paper":{"title":"Modules Satisfying the Prime Radical Condition and a Sheaf Construction for Modules I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RA"],"primary_cat":"math.AC","authors_text":"Mahmood Behboodi, Masoud Sabzevari","submitted_at":"2012-02-02T07:46:58Z","abstract_excerpt":"The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition \"$(\\sqrt[p]{{\\cal{P}}M}:M)={\\cal{P}}$\" for every prime ideal ${\\cal{P}}\\supseteq {\\rm Ann}(M)$, where $\\sqrt[p]{{\\cal{P}}M}$ is the intersection of all prime submodules of $M$ containing ${\\cal{P}}M$). This class contains the family of primeful modules properly. This yields that over any ring all free modules and all finitely generated modules lie in the class of $\\mathbb{P}$-radical modules. Also, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0377","kind":"arxiv","version":1},"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"}