I recently came across this problem during my study of modules: "Let R be a ring with unity and M a unital left R-module. For z$\in$M, let $_R$ denote the R-submodule of M generated by Z. Show $_RR$/$Ann_R(z) \cong_R$(z)$_R$, i.e. they are isomorphic as left R-modules."
My attempt so far has been to define $\phi$:$_RR \rightarrow$$_R$ by $\phi(r)=rz$. I've shown $\phi$ is an R-module homomorphism and that $ker(\phi)=Ann_R(z)$. I want to show that $\phi$ is onto so that I can use the First Isomorphism Theorem. However, with the definition I was given of (z)$_R$={rz+nz:r$\in$R and n$\in \mathbb{Z}$}, I am having trouble showing $\phi$ is onto.
Am I missing something trivial, or is my idea entirely wrong? Any guidance would be appreciated!