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2019 Minimal relations and catenary degrees in Krull monoids
Yushuang Fan, Alfred Geroldinger
J. Commut. Algebra 11(1): 29-47 (2019). DOI: 10.1216/JCA-2019-11-1-29

Abstract

Let $H$ be a Krull monoid with class group $G$. Then, $H$ is factorial if and only if $G$ is trivial. Sets of lengths and sets of catenary degrees are well studied invariants describing the arithmetic of $H$ in the non-factorial case. In this note, we focus on the set Ca $(H)$ of catenary degrees of $H$ and on the set $\mathcal R (H)$ of distances in minimal relations. We show that every finite nonempty subset of $\mathbb{N} _{\ge 2}$ can be realized as the set of catenary degrees of a Krull monoid with finite class group. This answers F. Halter-Koch [Problem 4.1]{N-P-T-W16a}. Suppose, in addition, that every class of $G$ contains a prime divisor. Then, Ca $(H)\subset \mathcal R (H)$ and $\mathcal R (H)$ contain a long interval. Under a reasonable condition on the Davenport constant of $G$, $\mathcal R (H)$ coincides with this interval, and the maximum equals the catenary degree of $H$.

Citation

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Yushuang Fan. Alfred Geroldinger. "Minimal relations and catenary degrees in Krull monoids." J. Commut. Algebra 11 (1) 29 - 47, 2019. https://doi.org/10.1216/JCA-2019-11-1-29

Information

Published: 2019
First available in Project Euclid: 13 March 2019

zbMATH: 07037587
MathSciNet: MR3923367
Digital Object Identifier: 10.1216/JCA-2019-11-1-29

Subjects:
Primary: 13A05 , 13F05 , 20M13

Keywords: catenary degrees , Krull monoids , minimal relations , sets of distances , sets of lengths

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.11 • No. 1 • 2019
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