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

for some $r_1,\dots,r_n\in R$. Equivalently, there is a surjective homomorphism $R^n\to M$, so $M$ is a quotient of a free module of finite rank.

The case $n=1$ recovers cyclic modules . Finiteness hypotheses are central for structure theorems (e.g. over a PID) and for Noetherian conditions.

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.