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| {{mergeto|semisimple module|date=April 2013}}
| | Nice to satisfy you, my name is Refugia. The favorite hobby for my kids and me is to play baseball and I'm trying to make it a occupation. My day job is a librarian. Puerto Rico is where he and his spouse live.<br><br>my site ... [http://videoworld.com/user/SMaloney home std test] |
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| In [[ring theory]], a branch of mathematics, a '''semisimple algebra''' is an [[associative algebra|associative]] [[artinian ring|artinian]] algebra over a [[field (mathematics)|field]] which has trivial [[Jacobson radical]] (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite dimensional this is equivalent to saying that it can be expressed as a Cartesian product of [[simple algebra|simple subalgebras]].
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| ==Definition==
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| The [[Jacobson radical]] of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module. The radical contains all nilpotent ideals, and if the algebra is finite-dimensional, the radical itself is a nilpotent ideal. A finite dimensional algebra is then said to be ''semisimple'' if its radical contains only the zero element.
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| An algebra ''A'' is called ''simple'' if it has no proper ideals and ''A''<sup>2</sup> = {''ab'' | ''a'', ''b'' ∈ ''A''} ≠ {0}. As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra ''A'' are ''A'' and {0}. Thus if ''A'' is not nilpotent, then ''A'' is semisimple. Because ''A''<sup>2</sup> is an ideal of ''A'' and ''A'' is simple, ''A''<sup>2</sup> = ''A''. By induction, ''A<sup>n</sup>'' = ''A'' for every positive integer ''n'', i.e. ''A'' is not nilpotent.
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| Any self-adjoint subalgebra ''A'' of ''n'' × ''n'' matrices with complex entries is semisimple. Let Rad(''A'') be the radical of ''A''. Suppose a matrix ''M'' is in Rad(''A''). Then ''M*M'' lies in some nilpotent ideals of ''A'', therefore (''M*M'')''<sup>k</sup>'' = 0 for some positive integer ''k''. By positive-semidefiniteness of ''M*M'', this implies ''M*M'' = 0. So ''M x'' is the zero vector for all ''x'', i.e. ''M'' = 0.
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| If {''A<sub>i</sub>''} is a finite collection of simple algebras, then their Cartesian product ∏ ''A<sub>i</sub>'' is semisimple. If (''a<sub>i</sub>'') is an element of Rad(''A'') and ''e''<sub>1</sub> is the multiplicative identity in ''A''<sub>1</sub> (all simple algebras possess a multiplicative identity), then (''a''<sub>1</sub>, ''a''<sub>2</sub>, ...) · (''e''<sub>1</sub>, 0, ...) = (''a''<sub>1</sub>, 0..., 0) lies in some nilpotent ideal of ∏ ''A<sub>i</sub>''. This implies, for all ''b'' in ''A''<sub>1</sub>, ''a''<sub>1</sub>''b'' is nilpotent in ''A''<sub>1</sub>, i.e. ''a''<sub>1</sub> ∈ Rad(''A''<sub>1</sub>). So ''a''<sub>1</sub> = 0. Similarly, ''a<sub>i</sub>'' = 0 for all other ''i''.
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| It is less apparent from the definition that the converse of the above is also true, that is, any finite dimensional semisimple algebra is isomorphic to a Cartesian product of a finite number of simple algebras. The following is a semisimple algebra that appears not to be of this form. Let ''A'' be an algebra with Rad(''A'') ≠ ''A''. The quotient algebra ''B'' = ''A'' ⁄ Rad(''A'') is semisimple: If ''J'' is a nonzero nilpotent ideal in ''B'', then its preimage under the natural projection map is a nilpotent ideal in ''A'' which is strictly larger than Rad(''A''), a contradiction.
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| ==Characterization==
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| Let ''A'' be a finite dimensional semisimple algebra, and
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| :<math>\{0\} = J_0 \subset \cdots \subset J_n \subset A</math>
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| be a [[composition series]] of ''A'', then ''A'' is isomorphic to the following Cartesian product:
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| :<math>A \simeq J_1 \times J_2/J_1 \times J_3/J_2 \times ... \times J_n/ J_{n-1} \times A / J_n </math>
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| where each
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| :<math>J_{i+1}/J_i \,</math>
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| is a simple algebra.
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| The proof can be sketched as follows. First, invoking the assumption that ''A'' is semisimple, one can show that the ''J''<sub>1</sub> is a simple algebra (therefore unital). So ''J''<sub>1</sub> is a unital subalgebra and an ideal of ''J''<sub>2</sub>. Therefore one can decompose
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| :<math>J_2 \simeq J_1 \times J_2/J_1 .</math>
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| By maximality of ''J''<sub>1</sub> as an ideal in ''J''<sub>2</sub> and also the semisimplicity of ''A'', the algebra
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| :<math>J_2/J_1 \,</math>
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| is simple. Proceed by induction in similar fashion proves the claim. For example, ''J''<sub>3</sub> is the Cartesian product of simple algebras
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| :<math>J_3 \simeq J_2 \times J_3 / J_2 \simeq J_1 \times J_2/J_1 \times J_3 / J_2.</math>
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| The above result can be restated in a different way. For a semisimple algebra ''A'' = ''A''<sub>1</sub> ×...× ''A<sub>n</sub>'' expressed in terms of its simple factors, consider the units ''e<sub>i</sub>'' ∈ ''A<sub>i</sub>''. The elements ''E<sub>i</sub>'' = (0,...,''e<sub>i</sub>'',...,0) are [[idempotent element]]s in ''A'' and they lie in the center of ''A''. Furthermore, ''E<sub>i</sub> A'' = ''A<sub>i</sub>'', ''E<sub>i</sub>E<sub>j</sub>'' = 0 for ''i'' ≠ ''j'', and Σ ''E<sub>i</sub>'' = 1, the multiplicative identity in ''A''.
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| Therefore, for every semisimple algebra ''A'', there exists idempotents {''E<sub>i</sub>''} in the center of ''A'', such that
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| #''E<sub>i</sub>E<sub>j</sub>'' = 0 for ''i'' ≠ ''j'' (such a set of idempotents is called ''[[Idempotent_element#Types_of_ring_idempotents|central orthogonal]]''),
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| #Σ ''E<sub>i</sub>'' = 1,
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| #''A'' is isomorphic to the Cartesian product of simple algebras ''E''<sub>1</sub> ''A'' ×...× ''E<sub>n</sub> A''.
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| ==Classification==
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| The [[Artin–Wedderburn theorem]] completely classifies semisimple algebras: they are isomorphic to a product <math> \prod M_{n_i}(D_i) </math> where the <math> n_i </math> are some integers, the <math> D_i </math> are [[division ring]]s, and <math> M_{n_i}(D_i) </math> means the ring of <math> n_i \times n_i </math> matrices over <math> D_i</math>. This product is unique up to permutation of the factors.
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| ==References==
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| [http://www.encyclopediaofmath.org/index.php/Semi-simple_algebra Springer Encyclopedia of Mathematics] | |
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| {{DEFAULTSORT:Semisimple Algebra}}
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| [[Category:Algebras]]
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Nice to satisfy you, my name is Refugia. The favorite hobby for my kids and me is to play baseball and I'm trying to make it a occupation. My day job is a librarian. Puerto Rico is where he and his spouse live.
my site ... home std test