I am having problems understanding an example constructed by M. Ojanguren and R. Sridharan showing that over the polynomial ring in two variables over a division ring (which is not a field) there exists a stably free module which is not free.
Let $k$ be a division ring that is not a field and $R=k[x,y]$. It can be easily shown that there exists a stably free $R$-module $P$ for which $P\oplus R\cong R^2$. Further we can show that there is a (right) ideal $J$ which is isomorphic to $P$ generated by the intersection of two principal ideals of $R$.
My question is why we can conclude from the above that $J$ must be generated by two elements and why this means that $J$, resp. $P$ is not free.
Can anybody help me with this?