On Monoids and Domains Whose Monadic Submonoids Are Krull
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A submonoid S of a given monoid H is called monadic if it is a divisor-closed submonoid of H generated by one element (i.e., there is some (non-zero) b ∈ H such that S is the smallest divisor-closed submonoid of H such that b ∈ S). In this paper we study monoids and domains whose monadic submonoids are Krull monoids. These monoids resp. domains are called monadically Krull. Every Krull monoid is a monadically Krull monoid, but the converse is not true. We provide several types of counterexamples and present a few characterizations for monadically Krull monoids. Furthermore, we show that rings of integer-valued polynomials over factorial domains are monadically Krull. Finally, we investigate the connections between monadically Krull monoids and generalizations of SP-domains.
KeywordsMonadically Integer-valued Krull monoid Mori set SP-domain
2000 Mathematics Subject Classification.13A15 13F05 20M11 20M12
We want to thank A. Geroldinger, F. Halter-Koch, F. Kainrath and the referee for their comments and suggestions. This work was supported by the Austrian Science Fund FWF, Project Number P21576-N18.
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