Semisimple ring

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.

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:

For ${\displaystyle 3\Rightarrow 2}$, the starting idea is to find an irreducible submodule by picking any nonzero ${\displaystyle x\in M}$ and letting ${\displaystyle P}$ be a maximal submodule such that ${\displaystyle x\notin P}$. It can be shown that the complement of ${\displaystyle P}$ is irreducible.

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