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**Formal Definition**

For rings $R$ and $R'$, a map $\phi: R \rightarrow R'$ is a **homomorphism** if the following two conditions are satisfied for all $a, b \in R$:

- $\phi(a+b) = \phi(a) + \phi(b)$,
- $\phi(ab) = \phi(a)\phi(b)$.

**Informal Definition**

This is essentially the same definition for a homomorphism between two groups, except that we have two conditions that must be satisfied, one for each operation in the rings.

**Example(s)**

Let $F$ be the ring of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. For each $a\in \mathbb{R}$, we have the **evaluation homomorphism** $\phi_a: F \rightarrow \mathbb{R}$, where $\phi_a(f) = f(a)$ for $f \in F$.

**Non-example(s)**

Replace this text with non-examples

**Additional Comments**

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