The ring of dyaidc rationals is $R=\{\frac{a}{b}:a,b\in\mathbb{Z}, b=2^n, n\in\mathbb{N}\}$.
I want to be able to say that if we have a rational $\frac{a}{b}$ in which $a=2^{j}e$ where $e$ is odd and $b=2^i$ then $\frac{a}{b}$ is dyadic if and only if $j, because otherwise we could reduce to $\frac{a}{b}=2^{j-i}e$ and it would not be in R.
However, if we insist upon writing things in this "simplest form" then I don't see how we could have that $1 \in R$, i.e. that $R$ is a ring with identity.
I've been asked to discuss the units (among other things) in this ring, but I can't make sense of that task for the reason I laid out above.