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

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## 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**.