View source for Module (mathematics)

In [[mathematics]], a '''module''' is one of the fundamental [[Algebraic structure|algebraic structures]] used in [[abstract algebra]].

== Description ==

A module over a [[Ring (mathematics)|ring]] is a generalization of the notion of [[vector space]] over a [[Field (mathematics)|field]], wherein the corresponding [[Scalar (mathematics)|scalars]] are the elements of an arbitrary given [[Ring (mathematics)|ring]] (with identity) and a [[multiplication]] (on the left and/or on the right) is defined between elements of the ring and elements of the module.

Thus, a module, like a vector space, is an additive [[abelian group]]; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is [[Semigroup action|compatible]] with the ring multiplication.

Modules are very closely related to the [[representation theory]] of groups. They are also one of the central notions of [[commutative algebr]] and [[homological algebra]], and are used widely in [[algebraic geometry]] and [[algebraic topology]].

== See also ==

* [[Algebra (ring theory)]]
* [[Group ring]]
* [[Module (model theory)]]
* [[Module spectrum]]

== External links ==

* [https://en.wikipedia.org/wiki/Module_(mathematics) Module (mathematics)] @ Wikipedia.org 

[[Category:Mathematics]]
[[Category:Set theory]]