[[Ring]]
# Ring homomorphism

A **ring homomorphism** is a [[Morphism|morphism]] in [[Category of rings]],
that is to say a structure-preserving map between [[ring|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

3. $f(1_{A})=f(1_{B})$

Sometimes these are referred to as **unital ring homomorphisms**.

## Properties

- A ring homomorphism $\varphi \in \Ring(R,S)$ is [[Monomorphism|monic]] iff it is [[Surjectivity, injectivity, and bijectivity|injective]] iff $\ker f = \{ 0 \}$
- A ring [[epimorphism]] need not be [[Surjectivity, injectivity, and bijectivity|surjective]]
  - e.g. inclusion $\iota : \mathbb{Z} \hookrightarrow \mathbb{Q}$. If $\alpha_{1}$ and $\alpha_{2}$ agree on $\mathbb{Z}$ they agree everywhere.



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