{"paper":{"title":"Krull dimension of monomial ideals in polynomial rings with real exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Sean Sather-Wagstaff, Zechariah Andersen","submitted_at":"2013-05-23T15:37:27Z","abstract_excerpt":"We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring $R=A[\\mathbb{M}^1_{\\geq 0}\\times\\cdots\\times\\mathbb{M}^d_{\\geq 0}]$ where $\\mathbb{M}^1,\\ldots,\\mathbb{M}^d$ are non-zero additive subgroups of $\\mathbb{R}$. We prove that the set of non-zero finitely generated monomial ideals in $R$ has the structure of a metric space, and we prove that a version of Krull dimension for this setting is lower semicontinuous with respect to thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.5460","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"}