- Thread starter
- #1
- Feb 29, 2012
- 342
Good afternoon! I wasn't able to get the necessary grade in abstract algebra and now I'm redoing many exercises and I would like some correction. All help is appreciated! 
Here is the question:
Show that every homomorphism of a field to a ring is one-to-one or null.
Let $\phi: F \to R$ be a homomorphism from a field $F$ to a ring $R$. Assume that $\phi \neq 0$. We have that $\phi(0_F) = 0_R$. Suppose that $\phi (a) = \phi(b)$. Since $F$ is a field, we will have $\phi(a-b) = 0_R$ if and only if $a-b=0$ and therefore $a=b$.
Cheers!

Here is the question:
Show that every homomorphism of a field to a ring is one-to-one or null.
Let $\phi: F \to R$ be a homomorphism from a field $F$ to a ring $R$. Assume that $\phi \neq 0$. We have that $\phi(0_F) = 0_R$. Suppose that $\phi (a) = \phi(b)$. Since $F$ is a field, we will have $\phi(a-b) = 0_R$ if and only if $a-b=0$ and therefore $a=b$.
Cheers!
