Indecomposable module

Remarks

Equivalently, $M \ne 0$ and the only idempotents of the endomorphism ring $\End_R(M)$ are $0$ and $1$. Example: the $\Z$-module $\Z$ is indecomposable. The check: two nonzero submodules $a\Z$ and $b\Z$ both contain $ab \ne 0$, so their sum is never direct. Example: $\Z/p^n\Z$ is indecomposable for $p$ prime and $n \ge 1$, since its subgroups form a chain. Non-example: $\Z/6\Z = \set{0, 3} \oplus \set{0, 2, 4}$ fails clause 2. Every simple module is indecomposable, but $\Z/4\Z$ is indecomposable and not simple.