In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.

In mathematics, a finitely generated module is a module that has a finite generating set.

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers .

In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain can be uniquely decomposed in much the same way that integers have a prime factorization.

In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra over a field where there exists a finite set of elements of such that every element of can be expressed as a polynomial in , with coefficients in .