Let $D$ be a division ring with center $F$, and $J$ a non-zero two-sided ideal of $D[x]$. Is it true that $J \cap F[x] \neq 0$?
This is a question spawned from another problem I'm working on. And I don't yet know how to crack it yet. Can somebody give me any insight?
To get the ball rolling I prove this in the affirmative in the case of (Hamiltonian) quaternions. I suspect a similar argument would work for all finite dimensional division algebras over a number field because such a division algebra is known to be a cyclic division algebra (split by a finite cyclic extension $K$ of $F$), but haven't thought about it enough.
So let $D=\Bbb{H}$ when $F=\Bbb{R}$. Let us also fix a splitting field $K=\Bbb{C}=F[i]$. Recall that $D=K\oplus Kj$ with $j^2=-1$, $ij=-ji$ implying that $jzj^{-1}=\overline{z}$ for all $z\in K$. Also recall that $K$ is the centralizer of $i$, and $iq=-qi$ for all the elements $q\in Kj$.
Claim 1: The ideal $J$ necessarily contains non-zero polynomials from $K[x]$.
Proof. Let $p(x)=\sum_{i=0}^n q_i x^i\in J$ be non-zero. For all $i$, write $$q_i=z_i+w_ij$$ with $z_i,w_i\in K$. As $J$ is 2-sided, it follows that $$ ip(x)i^{-1}=\sum_{i=0}^n(z_i-w_ij)x^i\in J. $$ Therefore both $\sum_iz_ix^i$ and $\sum_iw_ix^i=\left(\sum_i w_ij x^i\right)j^{-1}$ are also elements of $J$. At least one of them is non-zero. QED.
Claim 2. The ideal $J$ contains non-zero polynomials from $F[x]$.
Proof. By the first claim there exists a non-zero polynomial $p(x)=\sum_i z_i x^i\in J$ where $z_i\in K$ for all $i$. Consequently also the polynomial $$ \overline{p}(x)=\sum_i\overline{z_i}x^i=jp(x)j^{-1}\in J. $$ It is well known (and easy to prove) that $p(x)\overline{p}(x)\in F[x]$. Furthermore, it is also in the ideal $J$ and non-zero. QED.
We could also use $p(x)+\overline{p}(x)$ and $i(p(x)-\overline{p}(x))$ in the latter proof. Don't know which (if either) will work in the case of a more general division algebra with a cyclic extension $K/F$ of number fields and an element $a\in F^*$ describing the necessary 2-cocycle, see Matt Emerton's post for the description of the resulting $D$.
