阿廷-韦德伯恩定理

历史

1908年，约瑟夫·韦德伯恩发表了题为《论超复数》（英语：On hypercomplex numbers）的论文。该文中的定理23指出：任意一个单代数都可以表示成一个本原代数（即可除代数）和单矩阵代数的直积。利用现在的术语，该定理可以表述为：任何单代数都同构于某个除环上的有限阶矩阵环。1926年，埃米尔·阿廷将这一结论推广至半单环。

In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that a(n Artinian) semisimple ring R is isomorphic to the product of finitely many 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, 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 (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that R is isomorphic to the product of finitely many ni-by-ni matrix rings Mₙᵢ(Dᵢ) over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i.

There is also a version of the Wedderburn–Artin theorem for algebras over a field k. 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.

Proof

There are various proofs of the Wedderburn–Artin theorem. A common modern one takes the following approach.

Suppose the ring R is semisimple. Then the right R -module R_(R) is isomorphic to a finite direct sum of simple modules (which are the same as minimal right ideals of R). Write this direct sum as （公式略）

where the Iᵢ are mutually nonisomorphic simple right R -modules, the ith one appearing with multiplicity nᵢ. This gives an isomorphism of endomorphism rings （公式略）

and we can identify End (Iᵢ^⊕ nᵢ) with a ring of matrices （公式略）

where the endomorphism ring End (Iᵢ) Iᵢ is a division ring by Schur's lemma, because Iᵢ is simple. Since （公式略） we conclude （公式略）

Here we used right modules because （公式略）; if we used left modules R would be isomorphic to the opposite algebra of End (_(R)R), but the proof would still go through. To see this proof in a larger context, see Decomposition of a module. For the proof of an important special case, see Simple Artinian ring.

Consequences

Since a finite-dimensional algebra over a field is Artinian, the Wedderburn–Artin theorem implies that every finite-dimensional simple algebra over a field is isomorphic to an n-by-n matrix ring over some finite-dimensional division algebra D over k, where both n and D are uniquely determined. This was shown by Joseph Wedderburn. Emil Artin later generalized this result to the case of simple left or right Artinian rings.

Since the only finite-dimensional division algebra over an algebraically closed field is the field itself, the Wedderburn–Artin theorem has strong consequences in this case. Let R be a semisimple ring that is a finite-dimensional algebra over an algebraically closed field k. Then R is a finite product ∏ ᵢ₌₁^(r)Mₙᵢ(k) where the nᵢ are positive integers and Mₙᵢ(k) is the algebra of nᵢ× nᵢ matrices over k.

Furthermore, 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: that is, division algebras over k whose center is k. It implies that any finite-dimensional central simple algebra over k is isomorphic to a matrix algebra Mₙ(D)

where D is a finite-dimensional central division algebra over k.