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{{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.
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