TITLE: How to prove that homomorphism from field to ring is injective or zero?
QUESTION [9 upvotes]: Possible Duplicate:
Abstract Algebra ring homomorphism 

$f:K\rightarrow R$ is a homomorphism from field to ring. To prove is that it is injective or zero.

REPLY [20 votes]: A ring homomorphism $f: R \to R'$ is injective if and only if its kernel is $\{0\}$. 
Also, the kernel of a ring homomorphism is an ideal. 
Now let $f: K \to R$ be a ring homomorphism from a field into a ring. The only ideals in a field $K$ are $\{0\}$ and $K$ itself. Hence $f$ can either have kernel $\{0\}$ or $K$ itself. In the former case, $f$ is injective, in the latter case it is the zero map.
Hope this helps.