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I'm having some trouble understanding the difference between ring homomorphisms and field homomorphisms. Both seem to have similar definitions, i.e., preservation of addition, multiplication, and the multiplicative identity.

Say, is every ring homomorphism between two fields $F$ and $K$, automatically a field homomorphism or is some extra condition required?

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    $\begingroup$ Yes, a ring homomorphism betwen fieles is always a field homomorphism $\endgroup$
    – Marcos
    Oct 23, 2021 at 17:43
  • $\begingroup$ @MarcosEscartínFerrer Oh, I see. Thanks! Suppose $F$ is a field while $K$ is a ring. And $f$ is a ring homomorphism from $F$ to $K$. Then $f(F)$ is a field, right? (Although the whole of $K$ may not be a field and every element in $K$ may not have an inverse element.) $\endgroup$
    – S.D.
    Oct 23, 2021 at 17:47
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    $\begingroup$ Yeah, for proving this you just only have to check that the elements have inverses, which is very easy from the properties of homomorphism. $\endgroup$
    – Marcos
    Oct 23, 2021 at 17:51
  • $\begingroup$ Actually, it is just a ring homomorphism, since every field is a ring. So it is just a special situation with a special ring, which is a field. So instead of "field homomorphism" one could just say ring homomorphism between fields. $\endgroup$ Oct 23, 2021 at 18:11
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    $\begingroup$ @SanchayanDutta A small correction: If $f$ is a ring homomorphism from $F$ to $K$, where $F$ is a field and $K$ is a nonzero ring, then $f(F)$ is a field. $\endgroup$ Oct 23, 2021 at 18:49

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Yes, a field homomorphism is simply a ring homomorphism between fields.

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