On the arithmetic of tame monoids with applications to Krull monoids and Mori domains |
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Authors: | Alfred Geroldinger Florian Kainrath |
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Institution: | Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria |
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Abstract: | Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b∈H, let ω(H,b) be the smallest N∈N0∪{∞} having the following property: if n∈N and a1,…,an∈H are such that b divides a1⋅…⋅an, then b already divides a subproduct of a1⋅…⋅an consisting of at most N factors. The monoid H is called tame if . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M. |
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Keywords: | 13A05 13F05 20M13 |
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