Homomorphism Of A Ring

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 $a, b \in R$:

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

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