Definition:Ring Homomorphism

Definition
Let $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ be rings.

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

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

Then $\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$ 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

 * Definition:Homomorphism (Abstract Algebra)
 * Definition:Group Homomorphism


 * Definition:Ring Epimorphism: a surjective ring homomorphism


 * Definition:Ring Monomorphism: an injective ring homomorphism


 * Definition:Ring Isomorphism: a bijective ring homomorphism


 * Definition:Ring Endomorphism: a ring homomorphism from a ring to itself


 * Definition:Ring Automorphism: a ring isomorphism from a ring to itself