Graded integral domains and Nagata rings, II

Abstract

Let $D$ be an integral domain with quotient field $K$, $X$ be an indeterminate over $D$, $K[X]$ be the polynomial ring over $K$, and $R= \{f \in K[X] \mid f(0) \in D\}$; so $R$ is a subring of $K[X]$ containing $D[X]$. For $f = a_0 + a_1X + \cdots + a_nX^n \in R$, let $C(f)$ be the ideal of $R$ generated by $a_0, a_1X, \dots , a_nX^n$ and $N(H) = \{g \in R \mid C(g)_v = R\}$. In this paper, we study two rings $R_{N(H)}$ and Kr$(R, v) = \{\frac{f}{g} \mid f, g \in R$, $g \neq 0$, and $C(f) \subseteq C(g)_v\}$. We then use these two rings to give some examples which show that the results of \cite{ac13} are the best generalizations of Nagata rings and Kronecker function rings to graded integral domains.