When the Nagata ring ${D(X)}$ is a sharp domain
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Let $D$ be an integral domain, $X$ be an indeterminate over $D$, $D[X]$ be the polynomial ring over $D$, and $D(X)$ be the Nagata ring of $D$. Let $[d]$ be the star operation on $D[X]$, which is an extension of the $d$-operation on $D$ as in [5, Theorem 2.3]. In this paper, we show that $D$ is a sharp domain if and only if $D[X]$ is a $[d]$-sharp domain, if and only if $D(X)$ is a sharp domain.
Abstract
Let $D$ be an integral domain, $X$ be an indeterminate over $D$, $D[X]$ be the polynomial ring over $D$, and $D(X)$ be the Nagata ring of $D$. Let $[d]$ be the star operation on $D[X]$, which is an extension of the $d$-operation on $D$ as in [5, Theorem 2.3]. In this paper, we show that $D$ is a sharp domain if and only if $D[X]$ is a $[d]$-sharp domain, if and only if $D(X)$ is a sharp domain.
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