Cyclic module

For a generator $m$, the map $R\to M$, $r\mapsto rm$, is surjective with kernel equal to the annihilator $\operatorname{ann}(m)$. Hence

as left $R$-modules. Conversely, every quotient of the left regular module $R$ is cyclic.

• As a $\mathbb Z$-module, $\mathbb Z/n\mathbb Z$ is generated by $1\bmod n$.
• Any principal left ideal $Ra$ is generated by $a$.
• The zero module is cyclic, generated by $0$.