properties of semisimple modules

Let R be a ring. Recall that R-module M is called semisimplePlanetmathPlanetmathPlanetmathPlanetmath iff M is a direct sumMathworldPlanetmath of simple module.

PropositionPlanetmathPlanetmath. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath for R-module M:

  1. 1.

    M is semisimple;

  2. 2.

    M is generated by its simple submodules;

  3. 3.

    for every submodule NM there exists a submodule NM such that M=NN.

Title properties of semisimple modules
Canonical name PropertiesOfSemisimpleModules
Date of creation 2013-03-22 18:53:27
Last modified on 2013-03-22 18:53:27
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 16D60