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In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see Semisimple representation. A module over a (not necessarily commutative) ring 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 semisimple; i.e., 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 proof of the equivalences, see . The most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand. Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules. Let A be an algebra over a field K. Then a left module M over A is said to be absolutely semisimple if, for any field extension F of K, F ⊗K M is a semisimple module over F ⊗K A. If M is semisimple and N is a submodule, then N and M/N are also semisimple. An arbitrary direct sum of semisimple modules is semisimple. A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero. A semisimple module M over a ring R can also be thought of as a ring homomorphism from R into the ring of abelian group endomorphisms of M.

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In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" , while indecomposable "not expressible as ". A direct sum of indecomposables is called completely decomposable; this is weaker than being semisimple, which is a direct sum of simple modules.

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