- Show that any field morphism is either 0 or injective.
Solution
Let \(\phi:F\to R\) be a ring homomorphism with \(F\) a field. Its kernel is an ideal of \(F\). A field has only the ideals \((0)\) and \(F\), so either \[ \ker\phi=(0) \] or \[ \ker\phi=F. \] In the first case \(\phi\) is injective. In the second case \(\phi(x)=0\) for every \(x\in F\), so \(\phi\) is the zero map. Thus every ring homomorphism from a field is either zero or injective.
If ring homomorphisms are required to preserve \(1\), the zero case is excluded whenever \(R\ne0\).