Characterization of fields by injective homormorphisms.

Question

I am working on the converse of this result. Basically I want to close the characterization as:

Given a commutative ring $R$ with unit. If every non trivial homomorphism from $R$ is a monomorphism then $R$ is a field.

I think it has to be simple but I don't see a clear manner to do it.

Answer 1

Beware, your characterization is only true for commutative rings with units.

Suppose $R$ is not a field. Then it has an ideal $I$ which is not $(0)$ nor $R$. Just check that $\pi : R \rightarrow R/I$ is not a monomorphism and $\pi$ is not trivial to prove your result.