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Given $F$ is a field, $R$ is a ring, and $\phi:F\to R$ is a surjective ring homomorphism. How do we prove that this makes $\phi$ is a bijection and $R$ is a field.

Simplest possible explanation is most appreciated! I am looking for an intuitive understanding.

  • 2
    What elements can be in the kernel of $\phi$?2011-11-28
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    Do you know what all of the ideals in $F$ are? Can you see why $R$ must be a field if $\phi$ is bijective? By the way, it is assumed that $R$ is not the zero ring for this to be true.2011-11-28

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