= Solution
The intersection $J=I\cap\mathbb Z$ is an additive subgroup of $\mathbb Z$, and $nj\in I\cap\mathbb Z$ for $n\in\mathbb Z$, $j\in J$. Thus it is an <ideal> of $\mathbb Z$, necessarily $d\mathbb Z$ for a unique $d\ge0$.
Because $d\in I$, one has $dR\subseteq I$. Conversely, if $a/b\in I$ with $b$ <coprime> to $30$, multiplication by $b\in R$ gives $a\in I\cap\mathbb Z=d\mathbb Z$. Write $a=dk$; then $a/b=d(k/b)\in dR$. Thus $I=dR$.
Every positive <integer> factors as $d=2^\alpha3^\beta5^\gamma u$ with $\gcd(u,30)=1$. The factor $u$ is a <unit>, so it does not change the <ideal>. The complete distinct list is
$$
\boxed{(0),\qquad (2^\alpha3^\beta5^\gamma)_R\quad(\alpha,\beta,\gamma\ge0).}
$$
They are distinct because the intersection of the displayed nonzero <ideal> with $\mathbb Z$ is $2^\alpha3^\beta5^\gamma\mathbb Z$: admissible denominators cannot cancel any of these <prime> powers. The triple $(0,0,0)$ gives $R$ itself.
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