$Rx$ is a simple module

Question

Let $M\neq 0$ be an $R-$module. If $M=Rx$ for each $0\neq x\in M$, why is M a simple module?

Answer 1

Let $L$ be a submodule of $M$; if $L\ne0$, then take $x\in L$, $x\ne 0$. Since $Rx=M$ by assumption, we have $M=Rx\subseteq L\subseteq M$, so $L=M$.

Therefore $M$ is simple.