Solution of $ax+xb=c$ in a division ring

Question

The equation $ax+xb=c$ in the quaternions skew field ($a,b,c,x \in \mathbb{H}$) has solution: $$ x= \left(|b|^2+2b_0a +a^2\right)^{-1} \left( ac +c \bar b\right) $$ Where $|b|,b_0,\bar b$ are respectively the module, the real part and the conjugate of $b$. Searching a similar formula for a generic division ring $\mathbb{A}$ I find that, if there exists $\bar b \in \mathbb{A}$ and $h,k \in \mathcal{C}(\mathbb{A})$, where $ \mathcal{C}(\mathbb{A})$ is the center of $\mathbb{A}$, such that: $$ \begin{cases} b \bar b=h\\ b+\bar b=k \end{cases} $$ we have the solution: $$ x=\left(a^2+ka+h \right)^{-1} \left( ac +c \bar b\right) $$ Is that the only possible formal solution of the given equation?

Answer 1

Consider the ring of differential operators in x. By this I mean differential operators of the form $$ \sum_{i=0}^N p_i(x) {d^n \over dx^n}, $$ where $p_i$ is a polynomial in $x$. Multiplication is given by composition of operators. As a ring, this is generated by $x$ and $y=d/dx$. This is not a division ring, but can be embedded in one.

$x$ and $y$ satisfy an equation, $$ yx - xy = 1. $$ This is an equation of your form, where $a = x$, $b = -x$, and $c=1$, but is not an example of family of solutions you mention.