Let $R=\{a\cdot2^n\mid a,n \in \mathbb{Z}\}$ be a subring or the field of rational numbers $\mathbb Q$.
i) What kind of elements are invertible in $R$?
ii) Prove that $R$ is a principal ideal ring, that is, every ideal in $R$ is principal.
For (i) I came to a conclusion that whenever $a=1$, the elements are invertible. Is it right? or it might be hold for any odd a? (I doubt as 1/a is an integer only when a=1..)
For the (ii) I assumed that $J$, which is not 0, is an ideal in $R$. Thus, there exists $c\cdot2^k$ in $J$… and $c$ should be the minimal positive odd integer? but how to show it?