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- #1

- Feb 29, 2012

- 342

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 $a=b$.

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