Bulletin of the
Korean Mathematical Society BKMS

Let $*$ be an $e.a.b.$ star operation on an integrally closed domain $D$, and let $Kr(D,*)$ be the Kronecker function ring of $D$. We show that if $D$ is a P$*$MD, then the mapping $D_{\alpha} \mapsto Kr(D_{\alpha}, v)$ is a bijection from the set $\{D_{\alpha}\}$ of $*$-linked overrings of $D$ into the set of overrings of $Kr(D,v)$. This is a generalization of \cite[Proposition 32.19]{gilmer} that if $D$ is a Pr\"ufer domain, then the mapping $D_{\alpha} \mapsto Kr(D_{\alpha}, b)$ is a one-to-one mapping from the set $\{D_{\alpha}\}$ of overrings of $D$ onto the set of overrings of $Kr(D,b)$.

Keywords: star operation, Pr\"ufer $*$-multiplication domain, Kronecker function ring, $*$-linked overring

MSC numbers: 13A15, 13F05, 13G05