Cyclic module - Imported Knowlpedia Content

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

M=Rm=\{rm:r\in R\}.

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

Equivalent characterizations

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).

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 $0$ ).