Suppose $F$ is a field and $R$ is a ring. The function $f\colon F\to R$ is a surjective homomorphism. Prove that $R$ is either the trivial ring, or $R$ is isomorphic to $F$.
Field Homomorphisms
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abstract-algebra
ring-theory
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0Consider the kernel. – 2012-11-16
1 Answers
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We have a ring homomorphism $f : F \to R$ here are the key facts:
- the inverse image of an ideal is an ideal.
- the only ideals of a field $F$ are $(0)$ and $F$.
- the kernel of a ring homomorphism is an ideal.
Take the inverse image of the kernel, if it's $(0)$ the rings are isomorphic. If it's $F$ the homomorphism is trivial.