Let $\displaystyle f:R \rightarrow R$ a non-trivial ring homomorphism.

Prove:

a) $\displaystyle f(Q) \subset $ and $\displaystyle f|_Q = id_Q $

b) if $\displaystyle a > b$ then $\displaystyle f(a) > f(b)$

c) $\displaystyle f $ is smooth. Deduce that $\displaystyle f=id_R$

Thanks!