In the abstract algebra, a part of mathematics, the proposition of Artin-Wedderburn is a classification for semi-simple rings. The assertion is that a semi-simple ring R for some integers n1 and some dividing rings D1, both of which are uniquely determined for index index i na, is isomorphic with a product of nine-by-nine matrix rings across dividing rings D1. In particular, some single left or right artificial ring isomorphic with an n-by-n matrix ring over a division ring D, where both n and D are unique.
As a direct result, Artin-Wedderburn's statement implies that each single ring that is end-dimensional over a division ring (single algebra) is a matrix ring. This is the original result of Joseph Wedderburn. Emil Artin has later generalized this result for the case of the Artificial Ring.
Note that if R is a finite-dimensional single algebra about a dividing ring E, D must not be contained in E. Matrix rings about the complex numbers are, for example, end-dimensional single algebraes about the real numbers.
Artin-Wedderburn's statement reduces classifying single rings over a sharing ring to classifying sharing rings that contain a given sharing ring. This in turn can be simplified again: The center of D must be a field K. Therefore, R is a K-algebra and has R K as a center. A finite-dimensional single algebra R is thus a central single algebra about K. The proposition of Artin-Wedderburn thus reduces the problem of classifying end-dimensional central single algebra's into problem of classifying sharing rings with a given center. Examples
Let R be the field of the real numbers, C the complex numbers field and H are the quaternary fields. Also see
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