Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $.

Question

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $.

Help me some hints.

Thank you in advance.

Answer 1

Define

$$\phi: R\to Rx\le M\;,\;\;\phi(r):=rx$$

prove the above is a (left) $\;R$- module homomorphism, and now use the first isomorphism theorem.