f.g.

Adjective

finitely generated (not comparable)

1. (most generally) Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
2. (less generally, but still covering most cases outside pure category theory) Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set.
3. (abstract, usually, module, ring) Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)
4. (ring theory) Finitely generated as a (left) module over R.
5. Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).