nLab
finitely generated object

Definition

An object $A$ of a concrete category $C$ is finitely generated if it is a quotient object (in the sense of a regular epimorphism) of some free object $F$ in $C$, where $F$ is free on a finite set.

The object $A$ is finitely presented if it is the coequalizer of a parallel pair $R\rightrightarrows F$ such that $R$ is also free on a finite set.

See discussion at finitely presentable object more a more abstract version of these. See also generators and relations.

Examples

• For $R$ a ring, an $R$-module $N$ is finitely generated if it is a free module with a finite basis.