Definition:Ring Homomorphism

Definition

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 often define a ring homomorphism as a unital ring homomorphism.

Also known as

A ring homomorphism is also known as a (ring) representation.

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.