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]] | ||
[[Category:Module theory]] | [[Category:Module theory]] | ||
Revision as of 20:25, 4 February 2009
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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