{"paper":{"title":"Continuous closure, axes closure, and natural closure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.AC","authors_text":"Melvin Hochster, Neil Epstein","submitted_at":"2011-06-17T12:18:47Z","abstract_excerpt":"Let $R$ be a reduced affine $\\mathbb C$-algebra, with corresponding affine algebraic set $X$. Let $\\mathcal C(X)$ be the ring of continuous (Euclidean topology) $\\mathbb C$-valued functions on $X$. Brenner defined the \\emph{continuous closure} $I^{\\rm cont}$ of an ideal $I$ as $I\\mathcal C(X) \\cap R$. He also introduced an algebraic notion of \\emph{axes closure} $I^{\\rm ax}$ that always contains $I^{\\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \\in I^{\\rm ax}$ if its image is in $IS$ for every homomorphism $R \\to S$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3462","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"}