High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In ring theory, a semisimple algebra is an associative algebra 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. Given an algebra, its radical is the (unique) nilpotent ideal that contains all nilpotent elements in the algebra. A finite dimensional algebra is then said to be semi-simple if its radical is {0}, where 0 denotes the zero element of the algebra.

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