Usuario:Jgomez53/Esbozos3
De Wikipedia, la enciclopedia libre
El teorema de Wedderburn-Artin establece que un anillo semisimple A es isomorfo a un producto de k anillos de matrices de orden ni sobre anillos de división Ci donde k, n_i y Ci están determinados de forma única salvo el orden (i=1, 2, ..., k). Como consecuencia se obtiene que cualquier anillo simple y artiniano por la izquierda (o por la derecha) es isomorfo a un anillo de matrices de orden n sobre un anillo de división.
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.
[editar] 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.
- 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.
[editar] See also
- Maschke's theorem
