Let $\mathbb{F}$ be a field , $R$ be a ring , $\phi:\mathbb{F} \to R$ be an homomorphism. Prove $\phi$ is an injective homomorphism.

Question

Let $\mathbb{F}$ be a field , $R$ be a ring , $\phi:\mathbb{F} \to R$ be an homomorphism.

Prove $\phi$ is an injective homomorphism.

Suppose $\phi$ is not injective , hence $\exists a,b\in \mathbb{F}$ such that $\phi(a)=\phi(b) \implies \phi(a)-\phi(b)=0\implies \phi(a-b)=0$.

I have to conclude the $a-b=0$ , I have no idea how to approach the problem.

Appreciate any help.

Answer 1

This isn’t true. We could have $R$ be the zero ring.

However, if $R$ is not the zero ring, then this is true. For if $f(a - b) = 0$, then $f(a - b)$ is not a unit, and thus $a - b$ is not a unit. In a field, the only non-unit is zero, so $a - b = 0$ and thus $a = b$.