Definition:Ring Homomorphism

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

Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$.

That is, $\forall a, b \in R$:

\(\text {(1)}: \quad\) \(\ds \map \phi {a + b}\) \(=\) \(\ds \map \phi a \oplus \map \phi b\)
\(\text {(2)}: \quad\) \(\ds \map \phi {a \circ b}\) \(=\) \(\ds \map \phi a * \map \phi b\)

Then $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is a ring homomorphism.

Also defined as

Sources dealing only with rings with unity may define a ring homomorphism as a unital ring homomorphism.

Also known as

A ring homomorphism is also known as a (ring) representation, but that is reserved on $\mathsf{Pr} \infty \mathsf{fWiki}$ for a more specialised definition.

Some sources hyphenate: ring-homomorphism.

Also see

  • Results about ring homomorphisms can be found here.

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.