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Let $R, S$ be rings with unity.
Let $F$ be a subfield of both $R$ and $S$.
Then a ring homomorphism $\varphi: R \to S$ is called an $F$-homomorphism if:
- $\forall a \in F: \map \phi a = a$
That is, $\phi \restriction_F = I_F$ where:
- $\phi \restriction_F$ is the restriction of $\phi$ to $F$
- $I_F$ is the identity mapping on $F$.
The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.
Thus homomorphism means similar structure.