## You are here

Homehomomorphisms from fields are either injective or trivial

## Primary tabs

# homomorphisms from fields are either injective or trivial

Suppose $F$ is a field, $R$ is a ring, and $\phi\colon F\rightarrow R$ is a homomorphism of rings. Then $\phi$ is either trivial or injective.

###### Proof.

We use the fact that kernels of ring homomorphism are ideals. Since $F$ is a field, by the above result, we have that the kernel of $\phi$ is an ideal of the field $F$ and hence either empty or all of $F$. If the kernel is empty, then since a ring homomorphism is injective iff the kernel is trivial, we get that $\phi$ is injective. If the kernel is all of $F$, then $\phi$ is the zero map from $F$ to $R$. ∎

Type of Math Object:

Corollary

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

12E99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella