Editing Module (mathematics) (section)

{{Short description|Generalization of vector spaces from fields to rings}}
{{More footnotes|date=May 2015}}
{{Ring theory sidebar}}
{{Algebraic structures|module}}
In [[mathematics]], a '''module''' is a generalization of the notion of [[vector space]] in which the [[Field (mathematics)|field]] of [[scalar (mathematics)|scalars]] is replaced by a (not necessarily [[Commutative ring|commutative]]) [[Ring (mathematics)|ring]]. The concept of a ''module'' also generalizes the notion of an [[abelian group]], since the abelian groups are exactly the modules over the ring of [[integer]]s.<ref>Hungerford (1974) ''Algebra'', Springer, p 169: "Modules over a ring are a generalization of abelian groups (which are modules over Z)."</ref>

Like a vector space, a module is an additive abelian group, and scalar multiplication is [[Distributive property|distributive]] over the operations of addition between elements of the ring or module and is [[Semigroup action|compatible]] with the ring multiplication.

Modules are very closely related to the [[representation theory]] of [[group (mathematics)|group]]s. They are also one of the central notions of [[commutative algebra]] and [[homological algebra]], and are used widely in [[algebraic geometry]] and [[algebraic topology]].