TITLE: Finding the ideals in a ring of fractions
QUESTION [6 upvotes]: I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, $b$ is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find all the ideals of the given ring.
My initial thought is to find all the additive subgroups of $(R,+)$, but I am having trouble reasoning through this step. I'm not sure how to classify all such subgroups, then to prove that these are all such subgroups. 
A prod in the right direction would be greatly appreciated! ~Dom

REPLY [3 votes]: Step 1: Let $I$ be an ideal in $R$.  Show that $\mathfrak{i} = I \cap \mathbb{Z}$ is an ideal of $\mathbb{Z}$ and that $I = \{ \frac{i}{s} \ | \ i \in \mathfrak{i}, s \in \mathbb{Z}^+, \ 3 \nmid s \}$.  Deduce that if $\mathfrak{i} = n \mathbb{Z}$, then $I = n R$.  Thus all ideals of $R$ are principal and generated by elements of $\mathbb{Z}$.
Step 2: Figure out for which $m,n \in \mathbb{Z}^+$ we have $m R = nR$.  The answer given by Math Gems is relevant here.
Remark: $\S 7.2$ of these notes contains a more general discussion along these lines.  It amounts to finding the ideals in a localization, as Benjalim mentioned.