Every simple Module is Cyclic

Question

Can u help me prove that every simple module is cyclic? Ive already proved that a cyclic module is isomorphic to A quotiented by I ( I being a left ideal of A) using homomorphisms, but i cant seem to make any advance in proving this.Thx in advance.

Answer 1

Let M be a simple module. Then the only submodules of M are the trivial submodule and M itself. Suppose M is not cyclic, that is, it is not generated by any of its elements. Because M is a module, we know it is an additive subgroup and so the sets generated by each of its non-zero elements will form a subgroup of M but this contradicts our assumption that the only submodules of M are the trivial submodule and M itself

Answer 2

Suppose $M$ is a simple left $R$-module. This means that $M\ne\{0\}$ and that the only submodules of $M$ are $\{0\}$ and $M$.

Now take $x\in M$, $x\ne0$ (which exists by assumption). Then $Rx$ is a submodule of $R$ and $x=1x\in Rx$. Thus $Rx\ne\{0\}$ and so $Rx=M$.

---

If your rings don't necessarily have an identity, the definition of simple module is slightly different. A left $R$-module $M$ is simple if

1. $RM\ne\{0\}$;
2. the only submodules of $M$ are $\{0\}$ and $M$.

By assumption, there exist $r\in R$ and $x\in M$ such that $rx\ne0$. Now $Rrx$ is a nonzero submodule of $M$, so $Rrx=M$ and, a fortiori, $Rx=M$.