Let $D$ be a division ring. Let $D[x]$ be a polynomial ring in a central indeterminate $x$. And let $D(x)$ be the quotient ring of fraction of $D[x]$.
So, let $f,g \in D[x]$ such that $g \neq 0$ and $\operatorname{deg}(f) = \operatorname{deg}(g) = n$. Can we make sure that that there exists a pair of $s,t \in D[x]$ such that $\operatorname{deg}(s) \leq n$ and $f^{-1}g=st^{-1}$?
I'm new to non-commutative algebra, and I've just been working with basic stuffs. This one problem is a side project on my own where I get stuck.
I have an guts feeling it could be right, in the same way as talking about irreducible form of a fraction. But I don't really know. And I have no way to crack it. Any insight of telling me if this could be done or not, and some hint in case it could be done would be gladly appreciated.
Being new to noncommutative algebra, one thing you may not realize is going on i that "$f^{-1}g$ is a member of the ring of left quotients, and $st^{-1}$ is a member of the right ring of quotients.
It is known that in this case the two rings of quotients are isomorphic, but one needs to be cautious before beginning to equate their elements. Let's say that $\theta$ is the isomorphism from the right ring of quotients to the left ring of quotients.
Then $f^{-1}g=\theta(xy^{-1})=\theta(x)\theta(y)^{-1}$, so you could set $s=\theta(x)$ and $t=\theta(y)$ to indeed get $xt^{-1}$. But the wrinkle is that it is not immediately clear you have what you want in terms of the degrees of $f,g,s,t$. This indicates you would next consider whether or not $\theta$ restricted to $D[x]$ preserves degrees.
I hope this gets you started.