{"paper":{"title":"A note on Itoh (e)-Valuation Rings of and Ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"David E. Rush, Louis J. Ratliff, Youngsu Kim","submitted_at":"2016-07-18T22:31:04Z","abstract_excerpt":"Let $I$ be a regular proper ideal in a Noetherian ring $R$, let $e \\ge 2$ be an integer, let $\\mathbf T_e = R[u,tI,u^{\\frac{1}{e}}]' \\cap R[u^{\\frac{1}{e}},t^{\\frac{1}{e}}]$ (where $t$ is an indeterminate and $u =\\frac{1}{t}$), and let $\\mathbf r_e = u^{\\frac{1}{e}} \\mathbf T_e$. Then the Itoh (e)-valuation rings of $I$ are the rings $(\\mathbf T_e/z)_{(p/z)}$, where $p$ varies over the (height one) associated prime ideals of $\\mathbf r_e$ and $z$ is the (unique) minimal prime ideal in $\\mathbf T_e$ that is contained in $p$. We show, among other things:\n  (1) $\\mathbf r_e$ is a radical ideal if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05341","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"}