Editing Vector space (section)

===Modules===
{{Main|Module (mathematics)|l1=Module}}
''Modules'' are to [[ring (mathematics)|rings]] what vector spaces are to fields: the same axioms, applied to a ring ''R'' instead of a field ''F'', yield modules.{{sfn|Artin|1991|loc=ch. 12}} The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have [[multiplicative inverse]]s. For example, modules need not have bases, as the '''Z'''-module (that is, [[abelian group]]) [[Modular arithmetic|'''Z'''/2'''Z''']] shows; those modules that do (including all vector spaces) are known as [[free module]]s. Nevertheless, a vector space can be compactly defined as a [[Module (mathematics)|module]] over a [[Ring (mathematics)|ring]] which is a [[Field (mathematics)|field]], with the elements being called vectors.  Some authors use the term ''vector space'' to mean modules over a [[division ring]].{{sfn|Grillet|2007}} The algebro-geometric interpretation of commutative rings via their [[spectrum of a ring|spectrum]] allows the development of concepts such as [[locally free module]]s, the algebraic counterpart to vector bundles.