Definition:Antihomomorphism/Ring Antihomomorphism

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Definition

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.


Then $\phi: R \to S$ is a ring antihomomorphism if and only if:

\(\, \displaystyle \forall a, b \in R: \, \) \(\displaystyle \map \phi {a + b}\) \(=\) \(\displaystyle \map \phi a \oplus \map \phi b\)
\(\, \displaystyle \forall a, b \in R: \, \) \(\displaystyle \map \phi {a \circ b}\) \(=\) \(\displaystyle \map \phi b * \map \phi a\)