# Definition:Isomorphism (Abstract Algebra)/F-Isomorphism

## Definition

Let $R, S$ be rings with unity.

Let $F$ be a subfield of both $R$ and $S$.

Let $\phi: R \to S$ be an $F$-homomorphism such that $\phi$ is bijective.

Then $\phi$ is an $F$-isomorphism.

The relationship between $R$ and $S$ is denoted $R \cong_F S$.

## Linguistic Note

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.