Module (mathematics): Difference between revisions

Revision as of 12:29, 2 October 2024

Revision as of 14:35, 2 October 2024

Line 3:

{{Ring theory sidebar}}

{{Algebraic structures|module}}

In [[mathematics]], a '''module''' is a generalization of the notion of a [[vector space]] in which the [[Field (mathematics)|field]] of [[scalar (mathematics)|scalars]] is replaced by a (not necessarily [[Commutative_ring|commutative]]) [[Ring (mathematics)|ring]]. ForThe instance,concept anof abelian''module'' groupalso together withgeneralizes the operationnotion of adding[[abelian angroup]], elementsince tothe itselfabelian <math>n\in\mathbb{Z}</math>groups timesare formsexactly athe modulemodules butover notthe aring vectorof space[[integer]]s.

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.