Wedderburn–Artin theorem

In algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) 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.

Theorem

Let R be a semisimple ring. Then R is isomorphic to a product of finitely many n_i-by-n_i matrix rings $M_{n_{i}}(D_{i})$ over division rings D_i, for some integers n_i, both of which are uniquely determined up to permutation of the index i.

If R is a finite-dimensional semisimple k-algebra, then each D_i in the above statement is a finite-dimensional division algebra over k. The center of each D_i 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 Artin–Wedderburn 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 $k$ is an algebraically closed field, then the matrix ring having entries from $k$ is the only finite dimensional division algebra over $k$.

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 $\textstyle \prod _{i=1}^{r}M_{n_{i}}(k)$ where the $n_{i}$ are positive integers, and $M_{n_{i}}(k)$ is the algebra of $n_{i}\times n_{i}$ matrices over k.

Consequence

The Artin–Wedderburn 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

References

• 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. {{cite journal}}: Cite journal requires |journal= (help)