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Let $F$ be a field. Denote $R = \{ f/g \mid f, g \in F[x], g\neq 0 \}$, which is the fraction field of $F[x]$. Choose an element $a \in F$ and set $R_a = \{ f/g \in R\mid g(a) \neq 0 \}$.

Show that $R_a$ is a principal ideal domain, and describe all ideals of $R_a$.

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    $R_a$ is the localization of $F[X]$ at the maximal ideal $(X-a)$. This is a DVR and all its ideals are powers of the maximal ideal.2012-12-15

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