Ring homomorphism

A ring homomorphism is a structure-preserving map between rings. #m/def/ring Let $𝐴, 𝐵$ be rings and let $𝑓 : 𝐴 \to 𝐵$ . Then $𝑓$ is a ring homomorphism iff $𝑓$ is a rng homomorphism and in addition

$𝑓 (1 𝐴) = 𝑓 (1 𝐵)$

Sometimes these are referred to as unital ring homomorphisms. These are the morphisms in $𝖱 𝗂 𝗇 𝗀$ .

Properties

A ring homomorphism $𝜑 \in 𝖱 𝗂 𝗇 𝗀 (𝑅, 𝑆)$ is monic iff it is injective iff $k e r 𝑓 = {0}$
A ring epimorphism need not be surjective
- e.g. inclusion $𝜄 : ℤ ↪ ℚ$ . If $𝛼 1$ and $𝛼 2$ agree on $ℤ$ they agree everywhere.

#state/tidy | #lang/en | #SemBr