On transfer Krull monoids

Abstract

Let H be a cancellative commutative monoid, let mathcal {A}(H) be the set of atoms of H and let widetilde{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} is a DVM, then H is transfer Krull if and only if Hsubseteq widetilde{H} is inert. Moreover, we prove that if widetilde{H} is factorial, then H is transfer Krull if and only if mathcal {A}(widetilde{H})={uvarepsilon mid uin mathcal {A}(H),varepsilon in widetilde{H}^{times }}. We also show that if widetilde{H} is half-factorial, then H is transfer Krull if and only if mathcal {A}(H)subseteq mathcal {A}(widetilde{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.