Finitely generated module

Definition

Let $R$ be a ring with identity element and let $M$ be a left module over $R$. The module $M$ is finitely generated when there are $x_1, \dots, x_n \in M$ such that each $x \in M$ is $$x = r_1 x_1 + \cdots + r_n x_n \qquad \text{for some } r_1, \dots, r_n \in R.$$

Remarks

Over a field, the finitely generated modules are the finite-dimensional vector spaces. $\Z^n$ and $\Z / n\Z$ are finitely generated over $\Z$; the group $\Q$ is not. Over a Noetherian ring, each submodule of a finitely generated module is finitely generated. The finitely generated modules over a principal ideal domain are classified by the structure theorem.

Explanations

L1 human

A module is finitely generated when finitely many elements produce all others by linear combinations. This is the module analogue of a finite-dimensional vector space. Unlike in vector spaces, the generators need not be independent.