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单词 PropertiesOfSemisimpleModules
释义

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.

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