# Wedderburn–Artin theorem If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra over k. The center of each Di need not be k; it could be a finite extension of k.

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.

Corollary 1 The Wedderburn–Artin theorem implies that every simple ring that is finite-dimensional over a division ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined. This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of left or right Artinian rings. In particular, if {displaystyle k} is an algebraically closed field, then the matrix ring having entries from {displaystyle k} is the only finite dimensional Artinian simple algebra over {displaystyle k} .

Corollary 2 Let k be an algebraically closed field. Let R be a semisimple ring, that is a finite-dimensional k-algebra. Then R is a finite product {displaystyle textstyle prod _{i=1}^{r}M_{n_{i}}(k)} where the {displaystyle n_{i}} are positive integers, and {displaystyle M_{n_{i}}(k)} is the algebra of {displaystyle n_{i}times n_{i}} matrices over k.

Consequence The Wedderburn–Artin theorem reduces the problem of classifying finite-dimensional central simple algebras over a field K to the problem of classifying finite-dimensional central division algebras over K.

See also Maschke's theorem Brauer group Jacobson density theorem Hypercomplex number Emil Artin Joseph Wedderburn 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. ^ Jump up to: a b John A. Beachy (1999). Introductory Lectures on Rings and Modules. Cambridge University Press. p. 156. ISBN 978-0-521-64407-5. 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. Categories: Theorems in ring theory

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