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?