On the Cale Property in Integral Domains and Monoids

A monoid M is a Cale monoid with base Q if for every nonunit x ∈ M there exists a positive integer n such that xn factors uniquely up to order and associates as elements from Q ⊆ M\M. An integral domain D is a Cale domain with base Q, if its multiplicative monoid of nonzero elements is a Cale monoid with base Q. We explore the basic properties of Cale monoids and integral domains. In particular, we show that a Krull monoid is a Cale monoid if and only if its divisor class group is a torsion group. We show moreover that such Krull monoids arise exactly by taking the root closure of Cale monoids that satisfy a certain integral condition. Examples will also illustrate how the Cale property comes into play as a central arithmetical feature for monoids and domains that are not integrally closed.