Ring homomorphism

A ring homomorphism is a function $\varphi:R\to S$ between rings such that for all $a,b\in R$,

If $R,S$ are unital, one often additionally requires $\varphi(1_R)=1_S$ (a unital homomorphism).

Homomorphisms organize rings into a category; they compose via composition . Two fundamental invariants are the kernel and image, which control quotients and embeddings.

Examples:

• The reduction map $\mathbb Z\to \mathbb Z/n\mathbb Z$, $a\mapsto \overline a$, is a ring homomorphism.
• The inclusion $\mathbb Z\hookrightarrow \mathbb Q$ is a ring homomorphism.
• Evaluation at $c\in k$ gives a homomorphism $k[x]\to k$, $f\mapsto f(c)$.