on finite division subring of a ring 
Is there any example of a ring which is not a division ring but any of its  subring is a division ring? 

According to me if $R$ is a ring and $S$ is a division subring then $1\in S$ and hence $R=S$. Is it true?

A simple example is the product ring $\mathbb Z_2\times \Bbb Z_2$ (the Klein four group). Every proper nonzero subring is isomorphic to $\Bbb Z_2$, which is a division ring.
