Editing Linear algebra (section)

===Module theory===
{{main|Module (mathematics)}}

The existence of multiplicative inverses in fields is not involved in the axioms defining a vector space. One may thus replace the field of scalars by a [[ring (mathematics)|ring]] {{mvar|R}}, and this gives the structure called a '''module''' over {{mvar|R}}, or {{mvar|R}}-module.

The concepts of linear independence, span, basis, and linear maps (also called [[module homomorphism]]s) are defined for modules exactly as for vector spaces, with the essential difference that, if {{mvar|R}} is not a field, there are modules that do not have any basis. The modules that have a basis are the [[free module]]s, and those that are spanned by a finite set are the [[finitely generated module]]s. Module homomorphisms between finitely generated free modules may be represented by matrices. The theory of matrices over a ring is similar to that of matrices over a field, except that [[determinant]]s exist only if the ring is [[commutative ring|commutative]], and that a square matrix over a commutative ring is [[invertible matrix|invertible]] only if its determinant has a [[multiplicative inverse]] in the ring.

Vector spaces are completely characterized by their dimension (up to an isomorphism). In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules. However, every module is a [[cokernel]] of a homomorphism of free modules.

Modules over the integers can be identified with [[abelian group]]s, since the multiplication by an integer may be identified as a repeated addition. Most of the theory of abelian groups may be extended to modules over a [[principal ideal domain]]. In particular, over a principal ideal domain, every submodule of a free module is free, and the [[fundamental theorem of finitely generated abelian groups]] may be extended straightforwardly to finitely generated modules over a principal ring.

There are many rings for which there are algorithms for solving linear equations and systems of linear equations. However, these algorithms have generally a [[computational complexity]] that is much higher than similar algorithms over a field. For more details, see [[Linear equation over a ring]].