Question

Let K be a division ring with center k. Show that the center of the polynomial ring R = K[x] is k[x].

Answer 1

Clearly $k[x] \subseteq Z(R)$. Conversely, if $f = \sum a_{i}x^{i} \in Z(R)$, then $fa = af$ for all $a \in K$ shows that each $a_{-}i \in Z(K) = k$, and hence $f \in k[x]$.