On the complete integral closure of a domain

Abstract:

For a given positive integer n, a semivaluation domain ${D_n}$ is constructed so that the complete integral closure has to be applied successively exactly n times before obtaining a completely integrally closed domain. Letting ${G_n}$ be the group of divisibility of ${D_n}$, we set $G = \Sigma \boxplus {G_n}$, the cardinal sum of the groups ${G_n}$. It is concluded that the semivaluation domain D having G as its group of divisibility is a Bezout domain with the property that $D \subset {D^ \ast } \subset {D^{ \ast \ast }} \subset {D^{ \ast \ast \ast }} \subset \cdots$ is a strictly ascending infinite chain, where ${D^ \ast }$ is the complete integral closure of D.