Definition:Ring Homomorphism

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Definition

Let $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ be rings.

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

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

$(1):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \phi \left({a + b}\right)$	$=$	$\displaystyle \phi \left({a}\right) \oplus \phi \left({b}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(2):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \phi \left({a \circ b}\right)$	$=$	$\displaystyle \phi \left({a}\right) * \phi \left({b}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

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

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \phi \left({a + b}\right)\)	\(=\)	\(\displaystyle \phi \left({a}\right) \oplus \phi \left({b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \phi \left({a \circ b}\right)\)	\(=\)	\(\displaystyle \phi \left({a}\right) * \phi \left({b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)