Difference between revisions of "Module (mathematics)"
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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. | 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 | + | 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 == | == See also == | ||
Revision as of 12:08, 22 September 2016
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
Description
A module over a 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 (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 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
- Module (mathematics) @ Wikipedia.org
