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In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a 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 subalgebras. ==Definition== 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. An algebra ''A'' is called ''simple'' if it has no proper ideals and ''A''2 = ≠ . As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra ''A'' are ''A'' and . Thus if ''A'' is not nilpotent, then ''A'' is semisimple. Because ''A''2 is an ideal of ''A'' and ''A'' is simple, ''A''2 = ''A''. By induction, ''An'' = ''A'' for every positive integer ''n'', i.e. ''A'' is not nilpotent. 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'')''k'' = 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. If is a finite collection of simple algebras, then their Cartesian product ∏ ''Ai'' is semisimple. If (''ai'') is an element of Rad(''A'') and ''e''1 is the multiplicative identity in ''A''1 (all simple algebras possess a multiplicative identity), then (''a''1, ''a''2, ...) · (''e''1, 0, ...) = (''a''1, 0..., 0) lies in some nilpotent ideal of ∏ ''Ai''. This implies, for all ''b'' in ''A''1, ''a''1''b'' is nilpotent in ''A''1, i.e. ''a''1 ∈ Rad(''A''1). So ''a''1 = 0. Similarly, ''ai'' = 0 for all other ''i''. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Semisimple algebra」の詳細全文を読む スポンサード リンク
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