Additive commutators and the center of a division ring

Question

Can someone please explain me the following proof?

Proposition: If $D$ is a division ring, then the division ring $R$ generated by $Z(D)$ and all additive commutators (elements of the form $xy-yx$) is the whole of $D$.

Proof: Let $x$ be a member of $Z(D)$. Then there exists $y$ in $D$, such that $xy\neq yx$, therefore $x(xy - yx)$ is in $R^*$, and $xy - yx$ is in $R^*$, thus $x$ is in $R^*$.

Thanks! G.

Answer 1

Let $x\in D$. If $x\in Z(D)$, then $x\in R$. If $x\notin Z(D)$ then there is $y\in D$ such that $xy\neq yx$. But $xy-yx\in R^*$ and $x(xy-yx)=x(xy)-(xy)x\in R^*$, so $x\in R$.