Chapter

Commutative Algebra

pp 307-330

Date:

On Monoids and Domains Whose Monadic Submonoids Are Krull

  • Andreas ReinhartAffiliated withInstitut für Mathematik und wissenschaftliches Rechnen, Karl-Franzens-Universität Email author 

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Abstract

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) bH such that S is the smallest divisor-closed submonoid of H such that bS). 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.

Keywords

Monadically Integer-valued Krull monoid Mori set SP-domain

2000 Mathematics Subject Classification.

13A15 13F05 20M11 20M12