Category:Definitions/Ring Homomorphisms

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This category contains definitions related to Ring Homomorphisms.
Related results can be found in Category:Ring Homomorphisms.

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

Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$.

That is, $\forall a, b \in R$:

\(\text {(1)}: \quad\) \(\displaystyle \map \phi {a + b}\) \(=\) \(\displaystyle \map \phi a \oplus \map \phi b\)
\(\text {(2)}: \quad\) \(\displaystyle \map \phi {a \circ b}\) \(=\) \(\displaystyle \map \phi a * \map \phi b\)

Then $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is a ring homomorphism.