{"paper":{"title":"Generalized depth and associated primes in the perfect closure $R^\\infty$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"George Whelan","submitted_at":"2018-10-14T12:38:49Z","abstract_excerpt":"For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian. We first define the generalized notions of associated primes of a module over a non-Noetherian ring. Then for any $R$-module $M$, we state a correspondence between certain generalized prime ideals of $(R^\\infty \\otimes_R M)/N$ over $R^\\infty$, and the union of associated prime ideals of $F^e(M)/N_e$ as $e \\in \\mathbb{N}$ varies. Here $F$ refers to the Frobeni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06028","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"}