Artin-Wedderburn theorem

From Academic Kids

In abstract algebra, the Artin-Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that a semisimple ring R is isomorphic to a product of 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, this implies that 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.

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 by 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.
  • Every finite-dimensional simple algebra over C must be a matrix ring over C or H. 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.

See also

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