Difference between revisions of "Module"
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If <math>R</math> is a [[field]] then <math>M</math> is a vector space over <math>R</math>. | If <math>R</math> is a [[field]] then <math>M</math> is a vector space over <math>R</math>. | ||
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[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
[[Category:Ring theory]] | [[Category:Ring theory]] | ||
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Latest revision as of 19:11, 23 January 2017
A module is a type of object which appears frequently in abstract algebra. It is a generalization of the concept of a vector space.
Specifically, given a ring a (left) -module is an abelian group together with an operation (called scalar multiplication) written as or , which satisfies the following properties:
For all ,
(1)
(2)
(3)
(4)
We typically write to mean the module as well as the underlying abelian group.
If is a field then is a vector space over .
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