Arithmetics of flatness for monoids

Abstract

Flatness in categories of modules over rings has been transferred to categories of acts over semigroups by Stenström in 1971. In this note we will focus on commutative cancellative semigroups with unit (“monoids”) and show that this notion is useful too in considering arithmetical questions. The notion of factorable (resp. pre-Schreier) modules is extended to acts over monoids. For example it is shown that these acts can be characterized by flatness.

Keywords:

• Flatness
• integral domain
• overdomain

2020 Mathematics Subject Classification:

• 13A05
• 13A15
• 13B30
• 13C13
• 13G05

Acknowledgments

I thank the referee for his/her numerous and very valuable comments leading to a substantial improvement of this note, in particular concerning the last sections.