finitely generated module

A finitely generated module is one you can reach completely from a finite list of elements, using only addition and scalar multiplication by the ring — a module with a finite spanning set. It is the natural finiteness condition for modules: small enough to be controlled by finite data, but, unlike a free module, the generators need not be independent and there is usually no basis.

Precisely, an R-module M is finitely generated if there exist m_1, ..., m_n in M such that every element of M can be written as r_1 m_1 + ... + r_n m_n with r_i in R. Equivalently, M is a quotient of a free module of finite rank: there is a surjection R^n -> M. The relations among the generators (the kernel of this surjection) carry the rest of the information, and form what is called a presentation.

Being finitely generated is much weaker than being free or even Noetherian as a module in general — but over a Noetherian ring the two collapse: a module is finitely generated iff it is Noetherian, and then every submodule is finitely generated too. The whole structure theorem for modules over a PID is a statement about finitely generated modules, and Nakayama's lemma is the indispensable tool for reasoning about them locally.