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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 $F$.

Also see

Linguistic Note

The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.