Definition
A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.
For a module M, the following are equivalent:
- M is a direct sum of irreducible modules.
- M is the sum of its irreducible submodules.
- Every submodule of M is a direct summand: for every submodule N of M, there is a complement P such that M = N ⊕ P.
For, the starting idea is to find an irreducible submodule by picking any and letting be a maximal submodule such that . It can be shown that the complement of is irreducible.
Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules.
Read more about this topic: Semisimple Module
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