Wedderburn–Artin theorem explained

In algebra, the Wedderburn–Artin 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 -by- matrix rings over division rings, for some integers, both of which are uniquely determined up to permutation of the index . 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 be a (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that is isomorphic to a product of finitely many -by- matrix rings

M _{n _{i}} (D _{i)} over division rings, for some integers, both of which are uniquely determined up to permutation of the index .

There is also a version of the Wedderburn–Artin theorem for algebras over a field . If is a finite-dimensional semisimple -algebra, then each in the above statement is a finite-dimensional division algebra over . The center of each need not be ; it could be a finite extension of .

Note that if 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 asR _{R} \cong o plus ^{m} _{i =1} I ^{⊕ n _{i}} _{i} where the I _{i} are mutually nonisomorphic simple right R -modules, the th one appearing with multiplicity n _{i} . This gives an isomorphism of endomorphism ringsEnd (R _{R)} \cong o plus ^{m} _{i =1} End ( I ^{⊕ n _{i}} _{i} ) and we can identify End ( I ^{⊕ n _{i}} _{i} ) with a ring of matricesEnd ( I ^{⊕ n _{i}} _{i} ) \cong M _{n _{i}} ( End (I _{i))} where the endomorphism ring End (I _{i)} of I _{i} is a division ring by Schur's lemma, because I _{i} is simple. Since R \cong End (R _{R)} we concludeR \cong o plus ^{m} _{i =1} M _{n _{i}} ( End (I _{i))} .

Here we used right modules because

R \cong End (R _{R)} ; 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 be a semisimple ring that is a finite-dimensional algebra over an algebraically closed field

k . Then is a finite product style \prod ^{r} _{i =1} M _{n _{i}} (k) where the n _{i} are positive integers and M _{n _{i}} (k) is the algebra of n _{i} x n _{i} 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 style M _{n} (D) where D is a finite-dimensional central division algebra over k .

See also

• Maschke's theorem
• Brauer group
• Jacobson density theorem
• Hypercomplex number
• Emil Artin
• Joseph Wedderburn

References

• Book: Beachy . John A. . 1999 . Introductory Lectures on Rings and Modules . registration . Cambridge University Press . 978-0-521-64407-5 . 156 .
• Book: Cohn . P. M. . 2003 . Basic Algebra: Groups, Rings, and Fields . 137–139 .
• Henderson . D.W. . 1965 . A short proof of Wedderburn's theorem . . 72 . 4 . 385–386 . 10.2307/2313499 . 2313499 .
• Nicholson . William K. . 1993 . A short proof of the Wedderburn–Artin theorem . New Zealand J. Math . 22 . 83–86 .
• Wedderburn . J.H.M. . Joseph Wedderburn . 1908 . On Hypercomplex Numbers . . 6 . 77–118 . 10.1112/plms/s2-6.1.77 .
• Artin . E. . Emil Artin . 1927 . . 53.0114.03 . 10.1007/BF02952526 . Zur Theorie der hyperkomplexen Zahlen . 5 . 251–260 .