The complete integral closure of 𝑅[𝑋]

Abstract

For a reduced ring R R that is completely integrally closed it is not always the case that the corresponding polynomial ring R [ X ] R[X] is completely integrally closed. In this paper the question of when R [ X ] R[X] is completely integrally closed is shown to be related to the question of when R R is completely integrally closed in T ( R [ X ] ) T(R[X]) the total quotient ring of R [ X ] R[X] . A characterization of the complete integral closure of R [ X ] R[X] is given in the main theorem and this result is used to characterize the complete integral closure of the semigroup ring R [ S ] R[S] when S S is a torsion-free cancellative monoid.