Prime and maximal ideals of this ring? $R=\bigl\{\frac{a}{b}\  |  \gcd(3, b) = 1,\ a, b ∈ \Bbb Z\bigr\}$ 


I am trying to find all of the ideals in this ring. So far I have $\bigl\{0\bigr\}$, $R$, and $S=\bigl\{\frac{a}{b}\  |  \gcd(3, b) = 1, \gcd(3,a)=3 \ a, b ∈ \Bbb Z\bigr\}$ . I'm not sure how to find any more or, if there are no more, show that this is the case. I am then interested in which of the ideals are prime and maximal.
This is the ring of fractions with denominator not divisible by $3$. In other words, it is the localisation of $\mathbf Z$ at the prime ideal $3\mathbf Z$, usually denoted as $\mathbf Z_{(3)}$.

It is known the only non-trivial ideals of this ring are generated by the powers of $3$, so the only prime ideals are $(0)$ and $3R$ (which is the single maximal ideal of $R$).
This is the ring of fractions with denominator not divisible by $3$. In other words, it is the localisation of $\mathbf Z$ at the prime ideal $3\mathbf Z$, usually denoted as $\mathbf Z_{(3)}$.

It is known the only non-trivial ideals of this ring are generated by the powers of $3$, so the only prime ideals are $(0)$ and $3R$ (which is the single maximal ideal of $R$).