1. ## Ring Homomorphism

Prove that if F is a field, R is a ring, and $\displaystyle \theta :F\rightarrow R$ is a ring homomorphism, then either$\displaystyle \theta$ is one to one or $\displaystyle \theta(a)=0$ for all a in F.

i'm stuck on this proof.

Thank you.

2. Originally Posted by hercules
Prove that if F is a field, R is a ring, and $\displaystyle \theta :F\rightarrow R$ is a ring homomorphism, then either$\displaystyle \theta$ is one to one or $\displaystyle \theta(a)=0$ for all a in F.
Hint: $\displaystyle \ker \theta$ is an ideal of $\displaystyle F$.