Math 2.0

Associative rings and algebras Modules and ideals Theorem

Characterizations of semisimple modules

Builds on Module over a ringSubmoduleSimple moduleSemisimple moduleZorn's lemma

Statement

Let RR be a ring and let MM be a left RR-module. The following are equivalent:

  1. MM is semisimple, that is, a direct sum of simple submodules;
  2. MM is a sum of simple submodules;
  3. every submodule NN of MM is a direct summand: there is a submodule N′N' with M=N⊕N′M = N \oplus N'.

Remarks

The step from 2 to 1, and to 3, uses Zorn's lemma. Given M=∑i∈ISiM = \sum_{i \in I} S_i and a submodule NN, choose a set J⊆IJ \subseteq I maximal such that the sum N+∑j∈JSjN + \sum_{j \in J} S_j is direct. Each SiS_i meets this sum, and being simple it lies inside it, so the sum is all of MM; taking N=0N = 0 gives 1. For 3 implies 2, one shows that every nonzero submodule contains a simple submodule, and that the sum of all simple submodules has a complement which must be zero. Consequences: submodules and quotient modules of semisimple modules are semisimple. Example: in a vector space every subspace has a complementary subspace, matching condition 3.

Going deeper

For a student

Condition 3 is usually the easiest to check, and it is how Maschke's theorem proves semisimplicity of group representations. Condition 2 shows quickly that quotients of semisimple modules are semisimple. The equivalence is the starting point of the theory of semisimple rings, where every module satisfies these conditions.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionModule over a ring, definition, no evidence levelModule over a ringdefinitionProposition, concept, no evidence levelPropositionconceptSemisimple module, concept, no evidence levelSemisimple moduleconceptSimple module, concept, no evidence levelSimple moduleconceptSet, definition, no evidence levelSetdefinitionSubmodule, definition, no evidence levelSubmoduledefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionChoice function, definition, no evidence levelChoice functiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementMaximal element, definition, no evidence levelMaximal elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionUpper bound, definition, no evidence levelUpper bounddefinitionChain, definition, no evidence levelChaindefinitionZorn's lemma, statement, no evidence levelZorn's lemmastatementCharacterizations of semisimple modules, statement, no evidence levelCharacterizations of …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Module over a ring definition
  3. Proposition concept
  4. Semisimple module concept
  5. Simple module concept
  6. Set definition
  7. Submodule definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Family of sets definition
  12. Ordered pair definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Union of a family of sets definition
  18. Subset definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Antisymmetric relation definition
  22. Reflexive relation definition
  23. Transitive relation definition
  24. Function definition
  25. Preorder definition
  26. Choice function definition
  27. Partial order definition
  28. Axiom of choice statement
  29. Maximal element definition
  30. Total order definition
  31. Upper bound definition
  32. Chain definition
  33. Zorn's lemma statement
  34. Characterizations of semisimple modules statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Semisimple_module
About this page
TagE5D4BD4193
URIm2://E5D4BD4193
Kindstatement, theorem
WritingW0 stub
MSC16D60
StatusThis copy of the site has no build data. Run m2 status E5D4BD4193.
AboutSemisimple module
UsesModule over a ring, Submodule, Simple module, Semisimple module, Zorn's lemma

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