Krull dimension of monomial ideals in polynomial rings with real\n exponents

2013/05/23 by Zechariah Andersen, Andersen, Zechariah, Sean Sather-Wagstaff +1
Computer Science · Mathematics · #05C69 #05E40 #13F20 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Primary: 13C05 #Rings, Modules, and Algebras #Secondary: 05C22

paper · pdf · doi:10.48550/arxiv.1305.5460

openalex publication_date 2013/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop a new technique for studying monomial ideals in the standard\npolynomial rings A[X₁,\…,X_d] where A is a commutative ring with\nidentity. The main idea is to consider induced ideals in the semigroup ring\nR=A[ mathbbM¹_{\≥ 0}\×\⋯\× mathbbM^d_{\≥ 0}] where\n mathbbM¹,\…, mathbbM^d are non-zero additive subgroups of\n\ℝ. We prove that the set of non-zero finitely generated monomial\nideals in R has the structure of a metric space, and we prove that a version\nof Krull dimension for this setting is lower semicontinuous with respect to\nthis metric space structure. We also show how to use discrete techniques to\nstudy certain monomial ideals in this context.\n

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