[[Ring theory MOC]]
# Semisimple ring

A [[ring]] $R$ is **semisimple** iff it meets any of the following equivalent conditions: #m/def/ring

1. $R$ is [[Semisimple module|semisimple]] as a left [[module]]; ^SS1
2. $R$ is [[Semisimple module|semisimple]] as a right [[module]]; ^SS2
3. Every $R$-[[module]] is [[Semisimple module|semisimple]]; ^SS3
4. Every [[short exact sequence]] of $R$-[[module|modules]] [[Split short exact sequence|splits]]. ^SS4

> [!missing]- Proof of equivalence
> #missing/proof
> Right iff left follows from [[Wedderburn–Artin theorem]].

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