Editing Commutative ring (section)

=== Modules ===
{{Main|Module (mathematics)|l1=Module}}
For a ring <math> R </math>, an <math> R </math>-''module'' <math> M </math> is like what a vector space is to a field. That is, elements in a module can be added; they can be multiplied by elements of <math> R </math> subject to the same axioms as for a vector space.

The study of modules is significantly more involved than the one of [[vector space]]s, since there are modules that do not have any [[basis (linear algebra)|basis]], that is, do not contain a [[spanning set]] whose elements are [[linearly independent]]s. A module that has a basis is called a [[free module]], and a submodule of a free module needs not to be free.

A [[module of finite type]] is a module that has a finite spanning set. Modules of finite type play a fundamental role in the theory of commutative rings, similar to the role of the [[finite-dimensional vector space]]s in [[linear algebra]]. In particular, [[Noetherian rings]] (see also ''{{slink||Noetherian rings}}'', below) can be defined as the rings such that every submodule of a module of finite type is also of finite type.