Let $R \subset \mathbb{Q}$ be the subring $\left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \text { odd }\right\}$. Prove that the ideals of $R$ are the zero ideal $\{0\}$ and $2^{n} R$ for $n \geq 0$.
Hint: if $\{0\} \neq I \subset R$, the element $2^{n}\frac{\text { odd }}{\text { odd }}$ with the smallest $n$ generates $I$.
I have seen similar questions asked but I am stuck on the specific step which shows that $ I=2^{n} R $. I can show that there exists $n$ such that for every element $x$ of a non-zero ideal of R, it can be expressed as $x = 2^{n} \frac{a}{b}$ with $a, b \in \mathbb{Z}, b \text { odd }$. This gives me that $ I \subset 2^{n}R $, however I am having difficulties proving it in the other direction.
Apologies if this still counts as a duplicate question.