Editing Wedderburn–Artin theorem

{{Short description|Classification of semi-simple rings and algebras}}
In [[algebra]], the '''Wedderburn–Artin theorem''' is a [[classification theorem]] for [[semisimple ring]]s and [[semisimple algebra]]s.  The theorem states that an (Artinian){{efn|By the definition used here, [[semisimple ring]]s are automatically [[Artinian ring]]s. However, some authors use "semisimple" differently, to mean that the ring has a trivial [[Jacobson radical]]. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.}} semisimple ring ''R'' is isomorphic to a [[Product of rings|product]] of finitely many {{math|''n''{{sub|''i''}}}}-by-{{math|''n''{{sub|''i''}}}} [[matrix ring]]s over [[division ring]]s {{math|''D''{{sub|''i''}}}}, for some integers {{math|''n''{{sub|''i''}}}}, both of which are uniquely determined up to permutation of the index {{mvar|i}}.  In particular, any [[simple ring|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.{{sfn|Beachy|1999|ps=none}}

== Theorem ==
Let {{mvar|R}} be a (Artinian) [[semisimple ring]]. Then the Wedderburn–Artin theorem states that {{mvar|R}} is isomorphic to a product of finitely many {{math|''n''{{sub|''i''}}}}-by-{{math|''n''{{sub|''i''}}}} [[matrix ring]]s <math>M_{n_i}(D_i)</math> over [[division ring]]s {{math|''D''{{sub|''i''}}}}, for some integers {{math|''n''{{sub|''i''}}}}, both of which are uniquely determined up to permutation of the index {{mvar|i}}. 

There is also a version of the Wedderburn–Artin theorem for algebras over a [[Field (mathematics)|field]] {{mvar|k}}. If {{mvar|R}} is a finite-dimensional semisimple {{mvar|k}}-algebra, then each {{math|''D''{{sub|''i''}}}} in the above statement is a finite-dimensional [[division algebra]] over {{mvar|k}}.  The [[center (ring theory)|center]] of each {{math|''D''{{sub|''i''}}}} need not be {{mvar|k}}; it could be a [[finite field extension|finite extension]] of {{mvar|k}}.

Note that if {{mvar|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 number]]s are finite-dimensional simple algebras over the [[real number]]s.

== Proof ==
There are various [[Mathematical proof|proofs]] of the Wedderburn–Artin theorem.{{sfn|Henderson|1965|ps=none}}{{sfn|Nicholson|1993|ps=none}} A common modern one{{sfn|Cohn|2003|ps=none}} takes the following approach.  

Suppose the ring <math>R</math> is semisimple. Then the right <math>R</math>-module <math>R_R</math> is isomorphic to a finite direct sum of [[simple module]]s (which are the same as minimal right [[ideal (ring theory)|ideals]] of <math>R</math>).  Write this direct sum as
: <math>
   R_R  \;\cong\; \bigoplus_{i=1}^m I_i^{\oplus n_i}
</math>
where the <math>I_i</math> are mutually nonisomorphic simple right <math>R</math>-modules, the {{mvar|i}}th one appearing with multiplicity <math>n_i</math>.  This gives an isomorphism of [[endomorphism]] rings
: <math>
   \mathrm{End}(R_R) 
  \;\cong\; 
  \bigoplus_{i=1}^m \mathrm{End}\big(I_i^{\oplus n_i}\big) 
</math>
and we can identify <math>\mathrm{End}\big(I_i^{\oplus n_i}\big)</math> with a ring of [[matrix (mathematics)|matrices]]
: <math>
  \mathrm{End}\big(I_i^{\oplus n_i}\big) \;\cong\; M_{n_i}\big(\mathrm{End}(I_i)\big) 
</math>
where the endomorphism ring <math>\mathrm{End}(I_i)</math> of <math>I_i</math> is a division ring by [[Schur's lemma]], because <math>I_i</math> is simple.   Since <math>R \cong \mathrm{End}(R_R)</math> we conclude
: <math>
  R \;\cong\; 
  \bigoplus_{i=1}^m M_{n_i}\big(\mathrm{End}(I_i)\big) 
  \,.
</math>

Here we used right modules because <math>R \cong \mathrm{End}(R_R)</math>; if we used left modules <math>R</math> would be isomorphic to the [[opposite algebra]] of <math>\mathrm{End}({}_R R)</math>, 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 [[Artinian ring#Simple Artinian ring|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 <math> k </math>, where both ''n'' and ''D'' are uniquely determined.{{sfn|Beachy|1999|ps=none}} This was shown by [[Joseph Wedderburn]]. [[Emil Artin]] later generalized this result to the case of simple left or right [[Artinian ring]]s. 

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 {{mvar|R}} be a [[semisimple ring]] that is a finite-dimensional algebra over an algebraically closed field <math> k </math>.  Then {{mvar|R}} is a finite product <math>\textstyle \prod_{i=1}^r M_{n_i}(k) </math> where the <math> n_i </math> are positive integers and <math> M_{n_i}(k) </math> is the algebra of <math> n_i \times n_i </math> matrices over <math> k </math>.   

Furthermore, the Wedderburn–Artin theorem reduces the problem of classifying finite-dimensional [[central simple algebra]]s over a field <math> k </math> to the problem of classifying finite-dimensional '''central''' division algebras over <math> k </math>: that is, division algebras over <math> k </math> whose center is <math> k </math>.  It implies that any finite-dimensional central simple algebra over <math> k </math> is isomorphic to a matrix algebra <math>\textstyle M_{n}(D) </math>
where <math>D</math> is a finite-dimensional central division algebra over <math> k </math>.

== See also ==
* [[Maschke's theorem]]
* [[Brauer group]]
* [[Jacobson density theorem]]
* [[Hypercomplex number]]
* [[Emil Artin]]
* [[Joseph Wedderburn]]

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{cite book
  | last1=Beachy | first1=John A.
  | year=1999
  | title=Introductory Lectures on Rings and Modules
  | url=https://archive.org/details/introductorylect0000beac |url-access=registration
  | publisher=Cambridge University Press
  | isbn=978-0-521-64407-5
  | page=[https://archive.org/details/introductorylect0000beac/page/156 156]
  }}
* {{cite book
  | last1=Cohn | first1=P. M.
  | year=2003
  | title=Basic Algebra: Groups, Rings, and Fields
  | pages = 137–139
  }}
* {{cite journal
  | last1 = Henderson | first1 = D.W.
  | year = 1965
  | title = A short proof of Wedderburn's theorem
  | journal = [[The American Mathematical Monthly]]
  | volume = 72 | issue = 4
  | pages = 385–386
  | doi = 10.2307/2313499
  | jstor = 2313499
  }}
* {{cite journal
  | last1 = Nicholson | first1 = William K.
  | year = 1993
  | title = A short proof of the Wedderburn–Artin theorem
  | journal = New Zealand J. Math
  | volume = 22
  | pages = 83–86
  | url = https://www.thebookshelf.auckland.ac.nz/docs/NZJMaths/nzjmaths022/nzjmaths022-01-010.pdf
  }}
* {{cite journal
  | last1=Wedderburn | first1=J.H.M. | author-link=Joseph Wedderburn
  | year=1908
  | title=On Hypercomplex Numbers
  | journal=[[Proceedings of the London Mathematical Society]]
  | volume=6
  | pages=77–118
  | doi=10.1112/plms/s2-6.1.77
  | url=https://zenodo.org/record/1447798
  }}
* {{cite journal
  | last=Artin | first=E. | author-link = Emil Artin
  | year=1927
  | journal=[[Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg]]
  | jfm = 53.0114.03
  | doi = 10.1007/BF02952526
  | title=Zur Theorie der hyperkomplexen Zahlen
  | volume=5
  | pages=251–260
  }}
{{refend}}


{{DEFAULTSORT:Artin-Wedderburn Theorem}}
[[Category:Theorems in ring theory]]