I need to prove that if every left ideal of $R[x]$ is cyclic as a left $R[x]$-module then $R$ is a division ring.
I don't really understand. So a left ideal $I$ is cyclic as a left $R[x]$-module, let $f(x) \in R[x]$ we can write $I = R[x]f(x)$. $I$ is a left ideal and we can have $(R[x])^2f(x) \subset R[x]f(x)$. Then everything is already good. Where is the division ring part?