# Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism

(Redirected from Definition:Ring Isomorphism)
Jump to navigation Jump to search

## Definition

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.

Let $\phi: R \to S$ be a (ring) homomorphism.

Then $\phi$ is a ring isomorphism if and only if $\phi$ is a bijection.

That is, $\phi$ is a ring isomorphism if and only if $\phi$ is both a monomorphism and an epimorphism.

## 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.