Simple module

Remarks

Equivalently, $M \ne 0$ and $M = Rm$ for every nonzero $m \in M$. Example: for a prime $p$, the $\Z$-module $\Z/p\Z$ is simple. The check: its submodules are subgroups, whose orders divide $p$, so they are $0$ or $\Z/p\Z$. Example: for a field $K$, the column space $K^n$ is a simple $M_n(K)$-module, since any nonzero vector is sent to any other vector by some matrix. Non-example: the $\Z$-module $\Z$ is not simple, since $2\Z$ is a submodule different from $0$ and $\Z$; clause 2 fails. The zero module is not simple, by clause 1.