Homomorphism Of A Ring

Return to Glossary.

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

Additional Comments


Add any other comments you have about the term here

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License