Overrings of the Kronecker function ring $Kr(D, *)$ of a Pr\"ufer $*$-multiplication domain $D$
Bull. Korean Math. Soc. 2009 Vol. 46, No. 5, 1013-1018
https://doi.org/10.4134/BKMS.2009.46.5.1013
Printed September 1, 2009
Gyu Whan Chang
University of Incheon
Abstract : 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
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