KRONECKER FUNCTION RINGS AND PRUFER-LIKE DOMAINS

Abstract

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