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Definition (Ring Homomorphism And Isomorphism)
A ring homomorphism $\phi$ from a ring $R$ to a ring $S$ is a mapping from $R$ to $S$ that preserves the two ring operations. In other words $\phi(a+b)=\phi(a)+\phi(b)$ and $\phi(ab)=\phi(a)\phi(b)$. A bijective ring homomorphism is called a ring isomorphism.
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