Here is the problem I'm working on:

Let $\displaystyle R$ be an associative ring with $\displaystyle 1$, $\displaystyle M$ a right $\displaystyle R$-module, $\displaystyle F:M\rightarrow{R}$ a homomorphism of $\displaystyle R$-modules with $\displaystyle f(M)=R$. Prove that there is a decomposition $\displaystyle M=K\oplus{L}$ with $\displaystyle f(K)=0$ and $\displaystyle f|_L:L\rightarrow{R}$ is an isomorphism.

I think that $\displaystyle K$ should be the kernel of $\displaystyle F$. Then $\displaystyle L$ should be something like $\displaystyle M/K$, but it seems like I need something that's actually in $\displaystyle M$ instead. What should I use for $\displaystyle L$, and how do I show that $\displaystyle M=K\oplus{L}$ and $\displaystyle f|_L$ is an isomorphism?