# 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]}Also, the Artin–Wedderburn theorem says that a semisimple algebra that is finite-dimensional over a field [math]\displaystyle{ k }[/math] is isomorphic to a finite product [math]\displaystyle{ \prod M_{n_i}(D_i) }[/math] where the [math]\displaystyle{ n_i }[/math] are natural numbers, the [math]\displaystyle{ D_i }[/math] are finite dimensional division algebras over [math]\displaystyle{ k }[/math] (possibly finite extension fields of k), and [math]\displaystyle{ M_{n_i}(D_i) }[/math] is the algebra of [math]\displaystyle{ n_i \times n_i }[/math] matrices over [math]\displaystyle{ D_i }[/math]. Again, this product is unique up to permutation of the factors.

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.

## Consequence

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.

## See also

## 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. https://archive.org/details/introductorylect0000beac.

- 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**. pp. 251–260.