The ring $\mathbb{Z}[\frac{1}{2}]=\{\frac{u}{2^n}\}$ is the localization of $\mathbb{Z}$ in regards to $S=\{2^n:n \in \mathbb{N}\}$. Find all ideals of this ring.
I'm not sure what is meant by this question. The ideals should be of the form $(2^n u)$ since $\mathbb{Z}[\frac{1}{2}]$ has no other elements to form linear combinations?
Let $I$ be an ideal of $\mathbb{Z}[1/2]$, then $I\cap\mathbb{Z}$ is an ideal of $\mathbb{Z}$ so it is of the form $\alpha\mathbb{Z}$ with $\alpha\in\mathbb{N}$. I claim that $I=\alpha\mathbb{Z}[1/2]$. Indeed $\alpha\in I$ so $\alpha\mathbb{Z}[1/2]\subseteq I$, on the other hand if $x=\frac{a}{2^n}\in I$ then $2^nx=a\in I\cap\mathbb{Z}$ so $a=\alpha m$ for some $m\in\mathbb{Z}$ and therefore $x=\alpha\frac{m}{2^n}\in\alpha\mathbb{Z}[1/2]$.