Let R be a ring with identity. A left module M over R is a set with two binary operations, +:M×MM and :R×MM, such that

  1. 1.

    (𝐮+𝐯)+𝐰=𝐮+(𝐯+𝐰) for all 𝐮,𝐯,𝐰M

  2. 2.

    𝐮+𝐯=𝐯+𝐮 for all 𝐮,𝐯M

  3. 3.

    There exists an element 𝟎M such that 𝐮+𝟎=𝐮 for all 𝐮M

  4. 4.

    For any 𝐮M, there exists an element 𝐯M such that 𝐮+𝐯=𝟎

  5. 5.

    a(b𝐮)=(ab)𝐮 for all a,bR and 𝐮M

  6. 6.

    a(𝐮+𝐯)=(a𝐮)+(a𝐯) for all aR and 𝐮,𝐯M

  7. 7.

    (a+b)𝐮=(a𝐮)+(b𝐮) for all a,bR and 𝐮M

A left module M over R is called unitaryPlanetmathPlanetmath or unital if 1R𝐮=𝐮 for all 𝐮M.

A (unitary or unital) right module is defined analogously, except that the functionMathworldPlanetmath goes from M×R to M and the scalar multiplication operations act on the right. If R is commutativePlanetmathPlanetmathPlanetmath, 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