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WedderburnArtin theorem
If $R$ is a left semisimple ring, then
$R\cong\mathbb{M}_{{n_{1}}}(D_{1})\times\cdot\cdot\cdot\times\mathbb{M}_{{n_{r}% }}(D_{r})$ 
where each $D_{i}$ is a division ring and $\mathbb{M}_{{n_{i}}}(D_{i})$ is the matrix ring over $D_{i}$, $i=1,2,\ldots,r$. The positive integer $r$ is unique, and so are the division rings (up to permutation).
Some immediate consequences of this theorem:

A simple Artinian ring is isomorphic to a matrix ring over a division ring.

A commutative semisimple ring is a finite direct product of fields.
This theorem is a special case of the more general structure theorem on semiprimitive rings.
Related:
SemiprimitiveRing
Synonym:
structure theorem on semisimple rings, ArtinWedderburn theorem
Type of Math Object:
Theorem
Major Section:
Reference
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