Krull Dimension of Monomial Ideals in Polynomial Rings with Real Exponents

Abstract

We develop a new technique for studying monomial ideals in the standard polynomial rings A[X ₁,…, X _d ] where A is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring where 𝕄¹,…, 𝕄 ^d are nonzero additive subgroups of ℝ. We prove that the set of nonzero 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 this metric space structure. We also show how to use discrete techniques to study certain monomial ideals in this context.

Key Words:

• Edge ideals
• Krull dimension
• m-irreducible decompositions
• Monomial ideals
• Semicontinuous

2010 Mathematics Subject Classification:

• Primary: 13C05, 13F20
• Secondary: 05C22, 05E40, 05C69

ACKNOWLEDGMENTS

We are grateful Jon Totushek for teaching us how to create the diagrams for our examples, and to the anonymous referee for helpful suggestions.

Notes

It is worth noting that the ring has been studied previously, for instance, to construct interesting counterexamples to questions about non-noetherian rings; see, e.g., [1]. We are grateful to Jim Coykendall for teaching us about these constructions.

Despite this notation, note that 0 is not a monomial according to our definition.

Communicated by G. Leuschke.

To our teacher and friend, Jim Coykendall.