Ring homomorphism

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Flesh out, link to categorical concepts

Definition

Let $(R,+,\cdot)$ and $(S,\oplus,\odot)$ be rings and let $f:R\rightarrow S$ be a map. $f$ is a ring homomorphism (or just homomorphism, or morphism, if the context is clear) if:

1. $\forall a,b\in R[f(a+b)=f(a)\oplus f(b)]$ and
2. $\forall a,b\in R[f(a\cdot b)=f(a)\odot f(b)]$

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

• $f(0_R)=0_S$
• $\forall a\in R[f(-a)=-f(a)]$

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

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See also

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