## The complete integral closure of monoids and domains.(English)Zbl 0805.20055

In analogy with integral domains, the complete integral closure of a (multiplicative) commutative cancellative monoid $$H$$ with quotient group $$G$$ is defined to be $$\widehat{H} = \{x \in G \mid \text{ there is a }c \in H \text{ such that }cx^ n \in H \text{ for all } n \geq 1\}$$. This paper studies the complete integral closure for various classes of arithmetically defined monoids. These results are then applied to integral domains to recover several well known facts about the complete integral closure of an integral domain.

### MSC:

 20M14 Commutative semigroups 13G05 Integral domains 13B22 Integral closure of commutative rings and ideals