Module (mathematics): Difference between revisions

Revision as of 15:45, 18 September 2023

Revision as of 15:50, 18 September 2023

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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 [[Ring (mathematics)|ring]]. The concept of ''module'' elaboratesgeneralizes also the notion of [[abelian group]], insince that allthe abelian groups are exactly the modules over the ring of [[integer]]s but various abelian groups are modules over other rings as well.

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