Kronecker function rings and flat $D[X]$-modules

Abstract:

Let $D$ be an integral domain with identity. Gilmer has recently shown that in order that a $v$-domain $D$ be a Prüfer $v$-multiplication ring, it is necessary and sufficient that ${D^v}$ be a quotient ring of $D[X]$, where ${D^v}$ is the Kronecker function ring of $D$ with respect to the $v$-operation. In this paper the authors prove that in the above theorem it is possible to replace “a quotient ring of $D[X]$” with “a flat $D[X]$-module.” Moreover, it is shown that ${D^v}$ is the only Kronecker function ring of $D[X]$ which can ever be a flat $D[X]$-module.