Artin–Wedderburn theorem

In abstract 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_i matrix rings over division rings D_i, for some integers n_i, 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]

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 must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.
• Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C.
• Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.
• Every commutative semisimple ring must be a finite direct product of fields.^[3]
• The Artin–Wedderburn theorem implies that a semisimple algebra over a field $k$ is isomorphic to a finite product $\prod M_{n_i}(D_i)$ where the $n_i$ are natural numbers, the $D_i$ are finite dimensional division algebras over $k$, and $M_{n_i}(D_i)$ is the algebra of $n_i \times n_i$ matrices over $D_i$. Again, this product is unique up to permutation of the factors.

See also

• Maschke's theorem
• Brauer group
• Jacobson density theorem
• Hypercomplex number

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.
3. This is clear since matrix rings larger than 1×1 are never commutative.

• 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.