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AbstractFlatness 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. [ABSTRACT FROM AUTHOR]

It is shown that strong atoms are rather abundant in monadically Krull monoids. An application to rings of integer-valued polynomials on Krull domains yields new results on strong atoms in these rings. [ABSTRACT FROM AUTHOR]

It is shown that strong atoms are rather abundant in Krull monoids. For the proof extraction methods are used. An application of this result yields a positive answer to a question of D. D. Anderson, D. F. Anderson and J. Park about Dedekind domains. [ABSTRACT FROM AUTHOR]

In this note connections between root extensions of monoids and some finiteness conditions on monoids are studied, giving new proofs and generalizing results of Etingof, Malcolmson and Okoh for domains. In the same spirit, results of Jedrzejewicz and Zielinski on root-closed extensions of domains are generalized and sharpened to monoids. Using the same methods, a criterion for being a completely integrally closed domain is generalized to monoids. [ABSTRACT FROM AUTHOR]

A well-known generalization of the notion of primitive polynomials over factorial domains to arbitrary commutative rings is analyzed and applied to triangular systems. In particular, we clarify when a triangular system generates its saturation ideal, thus correcting a criterion published first in 2008. [ABSTRACT FROM AUTHOR]

Correction to: Semigroup Forum https://doi.org/10.1007/s00233-021-10231-9 This article was updated to remove two insertions and an error inadvertently introduced after the author had completed proofreading. The original article can be found online at https://doi.org/10.1007/s00233-021-10231-9. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. [Extracted from the article]