{"paper":{"title":"Iterated socles and integral dependence in regular rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alberto Corso, Bernd Ulrich, Claudia Polini, Craig Huneke, Shiro Goto","submitted_at":"2014-09-18T22:31:31Z","abstract_excerpt":"Let $R$ be a formal power series ring over a field, with maximal ideal $\\mathfrak m$, and let $I$ be an ideal of $R$ such that $R/I$ is Artinian. We study the iterated socles of $I$, that is the ideals which are defined as the largest ideal $J$ with $J\\mathfrak m^s\\subset I$ for a fixed positive integer $s$. We are interested in these ideals in connection with the notion of integral dependence of ideals. In this article we show that the iterated socles are integral over $I$, with reduction number one, provided $s \\leq \\text{o}(I_1(\\varphi_d))-1$, where $\\text{o}(I_1(\\varphi_d))$ is the order o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5481","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"}