On transfer Krull monoids

Abstract

AbstractLet H be a cancellative commutative monoid, let $$\mathcal {A}(H)$$ A ( H ) be the set of atoms of H and let $$\widetilde{H}$$ H ~ be the root closure of H. Then H is called transfer Krull if there exists a transfer homomorphism from H into a Krull monoid. It is well known that both half-factorial monoids and Krull monoids are transfer Krull monoids. In spite of many examples and counterexamples of transfer Krull monoids (that are neither Krull nor half-factorial), transfer Krull monoids have not been studied systematically (so far) as objects on their own. The main goal of the present paper is to attempt the first in-depth study of transfer Krull monoids. We investigate how the root closure of a monoid can affect the transfer Krull property and under what circumstances transfer Krull monoids have to be half-factorial or Krull. In particular, we show that if $$\widetilde{H}$$ H ~ is a DVM, then H is transfer Krull if and only if $$H\subseteq \widetilde{H}$$ H ⊆ H ~ is inert. Moreover, we prove that if $$\widetilde{H}$$ H ~ is factorial, then H is transfer Krull if and only if $$\mathcal {A}(\widetilde{H})=\{u\varepsilon \mid u\in \mathcal {A}(H),\varepsilon \in \widetilde{H}^{\times }\}$$ A ( H ~ ) = { u ε ∣ u ∈ A ( H ) , ε ∈ H ~ × } . We also show that if $$\widetilde{H}$$ H ~ is half-factorial, then H is transfer Krull if and only if $$\mathcal {A}(H)\subseteq \mathcal {A}(\widetilde{H})$$ A ( H ) ⊆ A ( H ~ ) . Finally, we point out that characterizing the transfer Krull property is more intricate for monoids whose root closure is Krull. This is done by providing a series of counterexamples involving reduced affine monoids.