homomorphisms from fields are either injective or trivial

Suppose F is a field, R is a ring, and ϕ:FR is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of rings. Then ϕ is either trivial or injectivePlanetmathPlanetmath.


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 ϕ 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 ϕ is injective. If the kernel is all of F, then ϕ is the zero map from F to R. ∎

Finally, it is clear that both of these possibilities are in fact achieved:

  • The map ϕ: given by ϕ(n)=0 is trivial (has all of as a kernel)

  • The inclusion [x] is injective (i.e. the kernel is trivial).

Title homomorphisms from fields are either injective or trivial
Canonical name HomomorphismsFromFieldsAreEitherInjectiveOrTrivial
Date of creation 2013-03-22 14:39:07
Last modified on 2013-03-22 14:39:07
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 4
Author mathcam (2727)
Entry type Corollary
Classification msc 12E99