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 see

 * Homomorphism
 * Group Homomorphism


 * Ring Epimorphism: a surjective ring homomorphism


 * Ring Monomorphism: an injective ring homomorphism


 * Ring Isomorphism: a bijective ring homomorphism


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


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