module
Let R be a ring with identity. A left module M over R is a set with two binary operations, +:M×M⟶M and ⋅:R×M⟶M, such that
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1.
(𝐮+𝐯)+𝐰=𝐮+(𝐯+𝐰) for all 𝐮,𝐯,𝐰∈M
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2.
𝐮+𝐯=𝐯+𝐮 for all 𝐮,𝐯∈M
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3.
There exists an element 𝟎∈M such that 𝐮+𝟎=𝐮 for all 𝐮∈M
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4.
For any 𝐮∈M, there exists an element 𝐯∈M such that 𝐮+𝐯=𝟎
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5.
a⋅(b⋅𝐮)=(a⋅b)⋅𝐮 for all a,b∈R and 𝐮∈M
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6.
a⋅(𝐮+𝐯)=(a⋅𝐮)+(a⋅𝐯) for all a∈R and 𝐮,𝐯∈M
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7.
(a+b)⋅𝐮=(a⋅𝐮)+(b⋅𝐮) for all a,b∈R and 𝐮∈M
A left module M over R is called unitary or unital if 1R⋅𝐮=𝐮 for all 𝐮∈M.
A (unitary or unital) right module is defined analogously, except that the function ⋅ goes from M×R to M and the scalar multiplication operations act on the right. If R is commutative
, there is an equivalence of categories between the category of left R–modules and the category of right R–modules.
Title | module |
Canonical name | Module |
Date of creation | 2013-03-22 11:49:14 |
Last modified on | 2013-03-22 11:49:14 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 11 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 13-00 |
Classification | msc 16-00 |
Classification | msc 20-00 |
Classification | msc 44A20 |
Classification | msc 33E20 |
Classification | msc 30D15 |
Synonym | left module |
Synonym | right module |
Related topic | MaximalIdeal |
Related topic | VectorSpace |