Module (mathematics) - Revision history

Karl Jones: /* See also */

2016-10-28T18:44:43Z

‎See also

Karl Jones at 18:08, 22 September 2016

2016-09-22T18:08:58Z

Karl Jones at 18:07, 22 September 2016

2016-09-22T18:07:17Z

Karl Jones: Created page with "In mathematics, a '''module''' is one of the fundamental algebraic structures used in abstract algebra. == Description == A module over a ring is..."

2016-09-22T18:06:01Z

Created page with "In mathematics, a '''module''' is one of the fundamental algebraic structures used in abstract algebra. == Description == A module over a ring is..."

New page

In [[mathematics]], a '''module''' is one of the fundamental 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, wherein the corresponding 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 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 algebra 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]]

@@ Line 5: / Line 5: @@
 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 compatible with the ring multiplication.
+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 algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.
+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 ==

@@ Line 14: / Line 14: @@
 * [[Group ring]]
 * [[Module (model theory)]]
-* [[Module spectrum
+* [[Module spectrum]]
 == External links ==

@@ Line 1: / Line 1: @@
-In [[mathematics]], a '''module''' is one of the fundamental algebraic structures used in abstract algebra.
+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, wherein the corresponding 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.
+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 compatible with the ring multiplication.