# Let F be a field and R be a ring. Suppose Φ:F-->R is an onto ring homomorphism. Prove R isomorphic to F or R is isomorphic to {0}.

SO since I have a field F I know the only ideals of F would be F itself and 0. I'm not sure how to tie this into the fact I have an onto ring homomorphism. i f you can let me know if I'm on the right track that'd be awesome

• A homomorphism out of a field is either injective or the 0 homomorphism. – Yeldarbskich Mar 14 '15 at 17:49

## 1 Answer

By what is given, $R\cong F/\ker\Phi$ and either $\ker\Phi=0$ or $\ker \Phi=F$.

• How do you know R is isomorphic to the factor ring F/kerΦ ? – Nicole Mar 14 '15 at 17:48
• 1st Isomorphism Theorem. – Nishant Mar 14 '15 at 17:48
• oh cool thanks! So I am missing where the onto ring homomorphism comes in to play – Nicole Mar 14 '15 at 17:49
• The first isomorphism theorem says that for $\phi : R\to S$ a ring homomorphism, $R/\ker\phi\cong im(\phi)$, but when $\phi$ is surjective, the image is all of $S$. – Stahl Mar 14 '15 at 17:52
• okay, so if I added Since F-->R is an onto ring homomorphism, we know kerΦ is an ideal of F. And then I continue my proof of how the only ideals of F are 0 and itself – Nicole Mar 14 '15 at 17:55