A finitely generated module over a ring 
is a module 
with a finite set of generators
,
..., 
such that every element of 
can be written as 
with 
. Equivalently, a left finitely generated module is
a quotient module of a finite-rank free
module 
(Bruns and Herzog 1998).
A Noetherian module is finitely generated, but the converse need not hold over an arbitrary ring. Over a Noetherian ring, every finitely generated module is a Noetherian module. Finite generation is the finiteness condition in the definition of a small Cohen-Macaulay module.
