# Thread: groups and rings

1. ## groups and rings

Show that if f and g are homomorphisms from Q into a ring R, Q the field of rational numbers, then if f(m) = g(m) for every integer m, f=g.

Any ideas?

2. ## Field of Fractions

Note $f(1)=1_R=g(1)$
$\forall n,m \in \mathbb{Z}$
$f(n)=f(n*1)=nf(1)=ng(1)=g(n)$
$f(1/m)=f(m^{-1})=f(m)^{-1}=g(m)^{-1}=g(m^{-1})=g(1/m)$
Thus for any $q/s \in \mathbb{Q}$
$f(q/s)=f(q*(1/s))=f(q)f(1/s)=g(q)g(1/s)=g(q/s)$
QED