Consider the ring $R := Q[t]_{(t)}=\{ f(t)/g(t) \mid g(0) ≠ 0\} ⊆ Q(t) = \operatorname{Frac}(Q[t])$. Then describe $R^×$, the invertible elements in $R$.
I know that Frac is a field and in this case it is a field relating to integers, however I don’t really understand how this can be related to finding the invertible elements in $R$.
“Frac” just denote the “field of fractions”, just like when you build $\mathbb{Q}$ from $\mathbb{Z}$, but here the construction is applied to $\mathbb{Q}[t]$ (the ring of polynomials over $\mathbb{Q}$).
The inverse of $f(t)/g(t)$ in $\mathbb{Q}(t)$ exists if and only if $f\ne0$ (not the zero polynomial) and is $g(t)/f(t)$. When does it belong to $\mathbb{Q}[t]_{(t)}$ according to the definition?