I've seen the following definition of semisimple algebra:
An algebra X is said to be semisimple if $\bigcap \{M \subset X : M \text{ is a maximal ideal}\} = \{0\}$.
But I couldn't find any examples of non-semisimple algebras in this context.
2026-03-27 20:53:59.1774644839
Examples of non-semisimple algebras
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Take an $n\times n$ matrix $A$ with complex entries. $(n>1)$.
Consider the collection of matrices obtained as the linear combination of all its powers $A^k, k\ge0$. (Cayley-Hamilton theorem assures us that we will not lose anything if we restrict ourselves upto $n$-th power.)
This is a subalgebra of matrix algebra. This subalgebra will be semisimple iff the chosen matrix $A$ is diagonalizable. (Actually this will be a commutative subalgebra).
(So diagonalizable matrices are also called semisimple matrices.)