# Artin–Wedderburn theorem

In algebra, the **Artin–Wedderburn theorem** is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) [1] semisimple ring *R* is isomorphic to a product of finitely many *n _{i}*-by-

*n*matrix rings over division rings

_{i}*D*, for some integers

_{i}*n*, both of which are uniquely determined up to permutation of the index

_{i}*i*. In particular, any simple left or right Artinian ring is isomorphic to an

*n*-by-

*n*matrix ring over a division ring

*D*, where both

*n*and

*D*are uniquely determined.[2]

As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a **simple algebra**) is a matrix ring. This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of Artinian rings.

Note that if *R* is a finite-dimensional simple algebra over a division ring *E*, *D* need not be contained in *E*. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.

The Artin–Wedderburn theorem reduces classifying simple rings over a division ring to classifying division rings that contain a given division ring. This in turn can be simplified: The center of *D* must be a field K. Therefore, *R* is a *K*-algebra, and itself has *K* as its center. A finite-dimensional simple algebra *R* is thus a central simple algebra over K. Thus the Artin–Wedderburn theorem reduces the problem of classifying finite-dimensional central simple algebras to the problem of classifying division rings with given center.

## Examples

Let **R** be the field of real numbers, **C** be the field of complex numbers, and **H** the quaternions.

- Every finite-dimensional simple algebra over
**R**is isomorphic to a matrix ring over**R**,**C**, or**H**. Every central simple algebra over**R**is isomorphic to a matrix ring over**R**or**H**. These results follow from the Frobenius theorem. - Every finite-dimensional simple algebra over
**C**is a central simple algebra, and is isomorphic to a matrix ring over**C**. - Every finite-dimensional central simple algebra over a finite field is isomorphic to a matrix ring over that field.
- For a commutative ring, the four following properties are equivalent: being a semisimple ring; being Artinian and reduced; being a reduced Noetherian ring of Krull dimension 0; being isomorphic to a finite direct product of fields.
- The Artin–Wedderburn theorem implies that a semisimple algebra that is finite-dimensional over a field is isomorphic to a finite product where the are natural numbers, the are finite dimensional division algebras over (possibly finite extension fields of k), and is the algebra of matrices over . Again, this product is unique up to permutation of the factors.

## References

- Semisimple rings are necessarily Artinian rings. Some authors use "semisimple" to mean the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.
- John A. Beachy (1999).
*Introductory Lectures on Rings and Modules*. Cambridge University Press. p. 156. ISBN 978-0-521-64407-5.

- P. M. Cohn (2003)
*Basic Algebra: Groups, Rings, and Fields*, pages 137–9. - J.H.M. Wedderburn (1908). "On Hypercomplex Numbers".
*Proceedings of the London Mathematical Society*.**6**: 77–118. doi:10.1112/plms/s2-6.1.77. - Artin, E. (1927). "Zur Theorie der hyperkomplexen Zahlen".
**5**: 251–260. Cite journal requires`|journal=`

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