Ring

Ring homomorphism

A ring homomorphism is a morphism in Category of rings, that is to say a structure-preserving map between rings. ring Let 𝐴,𝐵 be rings and let 𝑓 :𝐴 →𝐵. Then 𝑓 is a ring homomorphism iff 𝑓 is a rng homomorphism and in addition

3. 𝑓(1𝐴) =𝑓(1𝐵)

Sometimes these are referred to as unital ring homomorphisms.

Properties

• A ring homomorphism 𝜑 ∈𝖱𝗂𝗇𝗀(𝑅,𝑆) is monic iff it is injective iff ker⁡𝑓 ={0}
• A ring epimorphism need not be surjective
  • e.g. inclusion 𝜄 :ℤ ↪ℚ. If 𝛼1 and 𝛼2 agree on ℤ they agree everywhere.

---

tidy | en | SemBr