Ring homomorphism

If $R,S$ are unital, some authors additionally require $\varphi(1_R)=1_S$; this is then called a unital homomorphism. The definition above does not impose that condition, so it also applies to the nonunital rings allowed here.

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

• The inclusion $2\mathbb Z\hookrightarrow \mathbb Z$ is a ring homomorphism under the nonunital convention: neither the domain nor the map is required to preserve a multiplicative identity.

• 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 $k[x]\to k$, $f\mapsto f(c)$.