Finitely generated module

A left $R$ -module $M$ is finitely generated if there exist elements $m_1,\dots,m_n\in M$ such that every $m\in M$ can be written as

m=r_1m_1+\cdots+r_nm_n

for some $r_1,\dots,r_n\in R$ .

Finiteness hypotheses are central for structure theorems (e.g. over a PID) and for Noetherian conditions.

Remarks

The case $n=1$ recovers cyclic modules.

Equivalent characterizations

Equivalently, there is a surjective homomorphism $R^n\to M$ , so $M$ is a quotient of a free module of finite rank.

Examples

$\mathbb Z^n$ is finitely generated as a $\mathbb Z$ -module, generated by the standard basis vectors.
$\mathbb Z/12\mathbb Z$ is finitely generated (in fact cyclic) as a $\mathbb Z$ -module.
(Nonexample) $\bigoplus_{n\ge 1}\mathbb Z$ is not finitely generated as a $\mathbb Z$ -module.