KRONECKER FUNCTION RINGS AND PRUFER-LIKE DOMAINS

Abstract

Let $D$ be an integral domain, $D$ be the integral closure of $D$, $\ast$ be a star operation of finite character on $D$, ${\ast }_w$ be the so-called ${\ast }_w$-operation on $D$ induced by $\ast$, $X$ be an indeterminate over $D$, $N_{\ast }=f\in D[X]|c(f{)}^{\ast }=D$, and $Kr(D,\ast )=0\bigcup \frac{f}{g}|0=f,g\in D[X]\text{and there is an }0=h\in D[X]\text{such that }(c(f)c(h))\ast \subset (c(g)c(h){)}^{\ast }$. In this paper, we show that D is $a\ast$-quasi-Prufer domain if and only if $D[X{]}_{N_{\ast }}=Kr(D,{\ast }_w)$. As a
corollary, we recover Fontana-Jara-Santos’s result that D is a Prufer ∗-multiplication domain if and only if $D[X]N\ast =Kr(D,\ast w)$.