Proof that $F(a)=a+$ is an embedding from $K$ to $K[X]/$

I want to prove that when $$F:K\rightarrow K[X]/\langle f\rangle$$ is a map such that $$F(a)=a+\langle f \rangle$$, then $$F$$ is an embedding from $$K$$ to $$K[X]/\langle f \rangle$$, when $$f\in K[X]\backslash K$$.

To show the function is an embedding do I need to check that, for $$a,b\in K$$:

$$F(a+b)=F(a)+F(b)$$

Which seems straightforward in this case: $$F(a+b)=I+a+b=I+a+I+b=F(a)+F(b)$$

And the same for multiplication… And in addition check that the map is an injection?