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.

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.
↑ 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.

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Artin–Wedderburn theorem

Examples

See also

References