# Ring homomorphism

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Flesh out, link to categorical concepts

## Contents

## Definition

Let [ilmath](R,+,\cdot)[/ilmath] and [ilmath](S,\oplus,\odot)[/ilmath] be rings and let [ilmath]f:R\rightarrow S[/ilmath] be a map. [ilmath]f[/ilmath] is a *ring homomorphism* (or just *homomorphism*, or morphism, if the context is clear) if^{[1]}:

- [ilmath]\forall a,b\in R[f(a+b)=f(a)\oplus f(b)][/ilmath] and
- [ilmath]\forall a,b\in R[f(a\cdot b)=f(a)\odot f(b)][/ilmath]

As a consequence (see *immediate properties* below) we have:

- [ilmath]f(0_R)=0_S[/ilmath]
- [ilmath]\forall a\in R[f(-a)=-f(a)][/ilmath]

TODO: The case where [ilmath]R[/ilmath] has unity (is a u-ring), must [ilmath]S[/ilmath] then be too? We should have [ilmath]f(1_R)=1_S[/ilmath] so I guess if [ilmath]R[/ilmath] is a u-ring then [ilmath]S[/ilmath] must be too!

## See also

- Ring isomorphism
- An instance of isomorphism (category theory) - see isomorphism for a disambiguation.