Cyclic module

A left $R$-module $M$ is cyclic if there exists $m\in M$ such that

Equivalently, $M$ is a quotient of $R$ by the annihilator of the generator : the map $R\to M$, $r\mapsto rm$, is surjective with kernel $\operatorname{ann}(m)$, hence induces an isomorphism $R/\operatorname{ann}(m)\cong M$ (as in quotient modules ).

Cyclic modules are the building blocks for finitely generated modules and connect module structure to ideal structure in the ring.

Examples:

• As a $\mathbb Z$-module, $\mathbb Z/n\mathbb Z$ is cyclic generated by $1\bmod n$.
• Any principal ideal $(a)\subseteq R$ is a cyclic $R$-module generated by $a$.
• (Edge case) The zero module is cyclic (generated by $0$).