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
Those sources dealing only with rings with unity have the following additional condition for a ring homomorphism:


 * $\phi \left({1_R}\right) = 1_S$

where $1_R$ and $1_S$ are the unities of $R$ and $S$, respectively.

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