I just read in my linear algebra's notes the following statement : Let A be an integral domain and K a field. Any nonzero ring morphism $\phi : A \to K$ is injective.
I think this statement is false by considering the morphism $$\phi : \mathbb Z \to \mathbb Z /2 \mathbb Z$$ $$n \to [n]$$ This is a morphism between an integral domain and a field but clearly not injective.
So is the statement wrong ? I am quite sure of the counterexample but each time I disagreed with my teacher's notes, I was wrong.
You're correct. Here are two possibilities for what the statement should have been:
Any morphism from $K \to A$ is injective (because the kernel is an ideal of $K$ and the only ideals are $(0)$ and $(1) = K$). It doesn't matter so much that $A$ is an integral domain here other than to know that $A \neq 0$. If $A$ were $0$ then $K \to 0$ is non-injective.
The map $A \to \operatorname{Frac}(A)$ is injective.