Difference between revisions of "Module (mathematics)"
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| − | 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 == | == 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. | 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. | ||
Revision as of 11:07, 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 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
- Module (mathematics) @ Wikipedia.org
