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 ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index 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 $\displaystyle{ k }$ is isomorphic to a finite product $\displaystyle{ \prod M_{n_i}(D_i) }$ where the $\displaystyle{ n_i }$ are natural numbers, the $\displaystyle{ D_i }$ are finite dimensional division algebras over $\displaystyle{ k }$ (possibly finite extension fields of k), and $\displaystyle{ M_{n_i}(D_i) }$ is the algebra of $\displaystyle{ n_i \times n_i }$ matrices over $\displaystyle{ D_i }$. 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.

References

1. 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.
2. 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. pp. 251–260.