Could you please tell me whether my proof is correct or not,
$f(x)=f(x.1)=f(x).f(1)$ (Since we are finding ring homomorphism and unity belongs to set of Rationals, assuming f as ring homomorphism)
from above it is clear that $f(1)$ is unity element.
also $f(x) = f(1+1+1+1+.....+$x times) $=f(1)+f(1)+f(1)+.......+f(1) = x . f(1)$
or
$f(x) . f(1)= x . f(1) $
$f(1)(f(x)-x)=0$ (since Q is intergral domain, we get two cases)
$f(1) = 0$ corresponds to zero homomorphism
and $f(x) = x$ corresponds to Identity homomorphism
Thus we get only two ring homomorphisms from $f: \mathbb Q\to \mathbb Q$