# Definition:Antihomomorphism/Ring Antihomomorphism

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.
Then $\phi: R \to S$ is a ring antihomomorphism if and only if:
 $\, \displaystyle \forall a, b \in R: \,$ $\displaystyle \map \phi {a + b}$ $=$ $\displaystyle \map \phi a \oplus \map \phi b$ $\, \displaystyle \forall a, b \in R: \,$ $\displaystyle \map \phi {a \circ b}$ $=$ $\displaystyle \map \phi b * \map \phi a$