##  [Artin–Wedderburn Theorem](/artin-wedderburn-theorem) 

  ##  [Artin–Wedderburn Theorem](https://puremath.quantumdictionary.io/artin-wedderburn-theorem-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A classification theorem stating that any semisimple Artinian ring (equivalently any finite-dimensional semisimple algebra over a field) is isomorphic to a finite direct product of matrix algebras over division rings; in particular, simple artinian rings are matrix algebras over division rings.

 

 

 

 

 





 

 



 ##  [Artin–Wedderburn Theorem](https://algebra.quantumdictionary.io/artin-wedderburn-theorem-1) 

  

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- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The Artin–Wedderburn theorem classifies semisimple Artinian rings: any semisimple Artinian ring is isomorphic to a finite direct product of matrix algebras over division rings. In particular, a semisimple finite‑dimensional algebra over a field is a finite direct sum of full matrix algebras over division algebras.