Left Cyclic Modules and Canonical Homomorphism

Question

I am trying to show that if $_{R}M$ is a left R-module and cyclic, then $M\cong R/I$ for some left ideal $I$ in $R$.

If $l-ann(x)$ is the left annihilator of $x\in M$, then I showed that $l-ann(x)$ is a left ideal of $R$ and then I defined $\phi:R/l-ann(x)\to M$ by $\phi(r+l-ann(x))=(r+M)x=rx\in _{R}M\subset M$. But is this the correct mapping?

Answer 1

You know that $M$ has a generator $x$, which means $M=Rx$. Try defining a map $R\rightarrow M$ by $r\mapsto rx$. The kernel is exactly this annihilator you mention.